Topological spacesFrom Wikipedia, the free encyclopedia

Contents

1 a-paracompact space 11.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Adjunction space 22.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Categorical description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Appert topology 43.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Related topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4 Arens–Fort space 64.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

5 Box topology 75.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

5.2.1 Example - Failure at continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.2.2 Example - Failure at compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.2.3 Intuitive Description of Convergence; Comparisons . . . . . . . . . . . . . . . . . . . . . 8

5.3 Comparison with product topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

6 Cantor set 106.1 Construction and formula of the ternary set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

i

ii CONTENTS

6.2 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6.3.1 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116.3.2 Self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.3.3 Topological and analytical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.3.4 Measure and probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6.4 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.4.1 Smith–Volterra–Cantor set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.4.2 Cantor dust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6.5 Historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

7 Cantor space 167.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.2 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

8 Comb space 188.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.2 Topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

9 Compact convergence 219.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

10 Cosmic space 2310.1 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.2 Unsolved problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

11 CW complex 2411.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

CONTENTS iii

11.2 Inductive definition of CW complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2411.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2511.4 Homology and cohomology of CW complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2511.5 Modification of CW structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2611.6 'The' homotopy category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2711.7 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2711.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2711.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

11.9.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2711.9.2 General references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

12 Discrete space 2912.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2912.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2912.3 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112.4 Indiscrete spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

13 Discrete two-point space 3213.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

14 Dogbone space 3314.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3314.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

15 Dunce hat (topology) 3515.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3615.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

16 Equivariant topology 3716.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

17 Erdős space 3817.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

18 Euclidean space 3918.1 Intuitive overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3918.2 Euclidean structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

18.2.1 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4118.2.2 Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4218.2.3 Rotations and reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4218.2.4 Euclidean group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

18.3 Non-Cartesian coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

iv CONTENTS

18.4 Geometric shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4418.4.1 Lines, planes, and other subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4518.4.2 Line segments and triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4618.4.3 Polytopes and root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4718.4.4 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4718.4.5 Balls, spheres, and hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

18.5 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4818.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4818.7 Alternatives and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

18.7.1 Curved spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4818.7.2 Indefinite quadratic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4818.7.3 Other number fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4918.7.4 Infinite dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

18.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4918.9 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4918.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4918.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

19 Excluded point topology 5019.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5019.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

20 Extension topology 5120.1 Extension topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5120.2 Open extension topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5120.3 Closed extension topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5220.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

21 Finite topological space 5321.1 Topologies on a finite set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

21.1.1 As a bounded sublattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5321.1.2 Specialization preorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

21.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5421.2.1 0 or 1 points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5421.2.2 2 points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5421.2.3 3 points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

21.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5521.3.1 Compactness and countability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5521.3.2 Separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5521.3.3 Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5521.3.4 Additional structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5621.3.5 Algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

CONTENTS v

21.4 Number of topologies on a finite set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5621.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5721.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5721.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

22 First uncountable ordinal 5822.1 Topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5822.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5822.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

23 Fixed-point space 5923.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

24 Fort space 6024.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6024.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

25 Geometric topology (object) 6125.1 Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6125.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6125.3 Alternate definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

25.3.1 On framed manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6125.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6125.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

26 Half-disk topology 6326.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6326.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

27 Hausdorff space 6427.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6427.2 Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6527.3 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6527.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6527.5 Preregularity versus regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6627.6 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6627.7 Algebra of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6727.8 Academic humour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6727.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6727.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6727.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

28 Hawaiian earring 6828.1 Fundamental group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

vi CONTENTS

28.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

29 Hedgehog space 7029.1 Paris metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7029.2 Kowalsky’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7029.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7029.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7029.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

30 Hilbert cube 7230.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7230.2 The Hilbert cube as a metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7230.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7330.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7330.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

31 Hjalmar Ekdal topology 7431.1 Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7431.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

32 Homology sphere 7532.1 Poincaré homology sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

32.1.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7532.1.2 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

32.2 Constructions and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7632.3 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7632.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7732.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7732.6 Selected reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7732.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

33 Homotopy sphere 7833.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7833.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

34 Hyperbolic space 7934.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8034.2 Models of hyperbolic space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

34.2.1 The hyperboloid model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8034.2.2 The Klein model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8134.2.3 The Poincaré ball model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8134.2.4 The Poincaré half space model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

34.3 Hyperbolic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8234.3.1 Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

CONTENTS vii

34.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8234.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

35 Infinite broom 8335.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8435.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8435.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8435.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

36 Infinite loop space machine 8536.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

37 Interlocking interval topology 8637.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8637.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

38 Irrational winding of a torus 8738.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8738.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8738.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8738.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8838.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

39 K-topology 8939.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8939.2 Properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8939.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9039.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

40 Knaster–Kuratowski fan 9140.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9240.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

41 Lexicographic order topology on the unit square 9341.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9341.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9341.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9341.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

42 List of examples in general topology 9442.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

43 Long line (topology) 9643.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9643.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

viii CONTENTS

43.3 p-adic analog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9743.4 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9743.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9843.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

44 Loop space 9944.1 Relation between homotopy groups of a space and those of its loop space . . . . . . . . . . . . . . 9944.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9944.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

45 Lower limit topology 10145.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10145.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

46 Menger sponge 10346.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10346.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10546.3 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10646.4 MegaMenger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10646.5 Similar fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10646.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10746.7 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10746.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

47 Metric space 10947.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10947.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10947.3 Examples of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11047.4 Open and closed sets, topology and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 11147.5 Types of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

47.5.1 Complete spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11147.5.2 Bounded and totally bounded spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11247.5.3 Compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11347.5.4 Locally compact and proper spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11347.5.5 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11347.5.6 Separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

47.6 Types of maps between metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11347.6.1 Continuous maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11347.6.2 Uniformly continuous maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11447.6.3 Lipschitz-continuous maps and contractions . . . . . . . . . . . . . . . . . . . . . . . . . 11447.6.4 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11447.6.5 Quasi-isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

47.7 Notions of metric space equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

CONTENTS ix

47.8 Topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11547.9 Distance between points and sets; Hausdorff distance and Gromov metric . . . . . . . . . . . . . . 11547.10Product metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

47.10.1 Continuity of distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11647.11Quotient metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11647.12Generalizations of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

47.12.1 Metric spaces as enriched categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11747.13See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11747.14Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11847.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11947.16External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

48 Moore plane 12048.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12048.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12148.3 Proof that the Moore plane is not normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12148.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12148.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

49 Nilpotent space 12349.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

50 Overlapping interval topology 12450.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12450.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12450.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

51 Partially ordered space 12551.1 Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12551.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12551.3 External link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

52 Particular point topology 12652.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

52.1.1 Connectedness Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12652.1.2 Compactness Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12752.1.3 Limit related . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12752.1.4 Separation related . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

52.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12852.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

53 Partition topology 12953.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

x CONTENTS

54 Pointed space 13054.1 Category of pointed spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13054.2 Operations on pointed spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13054.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

55 Pointwise convergence 13255.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13255.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13255.3 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13355.4 Almost everywhere convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13355.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13355.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

56 Priestley space 13456.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13456.2 Properties of Priestley spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13456.3 Connection with spectral spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13556.4 Connection with bitopological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13556.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13556.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13656.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

57 Projectively extended real line 13757.1 Dividing by zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13757.2 Extensions of the real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13757.3 Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13757.4 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13857.5 Arithmetic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

57.5.1 Motivation for arithmetic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13957.5.2 Arithmetic operations that are defined . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13957.5.3 Arithmetic operations that are left undefined . . . . . . . . . . . . . . . . . . . . . . . . . 139

57.6 Algebraic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13957.7 Intervals and topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14057.8 Interval arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14057.9 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

57.9.1 Neighbourhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14057.9.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14157.9.3 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

57.10As a projective range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14257.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14257.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

58 Prüfer manifold 144

CONTENTS xi

58.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14458.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14458.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

59 Pseudocircle 14559.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

60 Pseudomanifold 14660.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

60.1.1 Implications of the definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14760.2 Related definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14760.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14760.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

61 Ran space 14861.1 Topological chiral homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14861.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14861.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14861.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

62 Real line 15062.1 As a linear continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15062.2 As a metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15162.3 As a topological space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15262.4 As a vector space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15362.5 As a measure space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15362.6 In real algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15362.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15462.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

63 Rose (topology) 15563.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15663.2 Relation to free groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15663.3 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15663.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15663.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

64 Shrinking space 15964.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

65 Sierpinski carpet 16065.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

65.1.1 Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16165.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

xii CONTENTS

65.3 Brownian motion on the Sierpinski carpet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16265.4 Wallis sieve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16265.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16265.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16365.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16365.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

66 Sierpinski triangle 16466.1 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

66.1.1 Removing triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16566.1.2 Shrinking and duplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16666.1.3 Chaos game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16766.1.4 Arrowhead curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16766.1.5 Cellular automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16866.1.6 Pascal’s triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16866.1.7 Towers of Hanoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

66.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16966.3 Generalization to other Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17066.4 Analogues in higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17166.5 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17266.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17266.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17266.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

67 Sierpiński space 17467.1 Definition and fundamental properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17467.2 Topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

67.2.1 Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17567.2.2 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17567.2.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17567.2.4 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17567.2.5 Metrizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17567.2.6 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

67.3 Continuous functions to the Sierpiński space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17667.3.1 Categorical description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17667.3.2 The initial topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

67.4 In algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17767.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17767.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17767.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

68 Simplicial complex 178

CONTENTS xiii

68.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17968.2 Closure, star, and link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17968.3 Algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17968.4 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18068.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18068.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18068.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

69 Simplicial space 18269.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

70 Smith–Volterra–Cantor set 18370.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18370.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18470.3 Other fat Cantor sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18470.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18470.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18470.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

71 Sorgenfrey plane 18571.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

72 Split interval 18772.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18772.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

73 Topological monoid 18873.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18873.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18873.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

74 Topological space 18974.1 History of Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18974.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

74.2.1 Neighbourhoods definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18974.2.2 Open sets definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19074.2.3 Closed sets definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19174.2.4 Other definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

74.3 Comparison of topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19174.4 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19274.5 Examples of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19274.6 Topological constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19374.7 Classification of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19374.8 Topological spaces with algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

xiv CONTENTS

74.9 Topological spaces with order structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19374.10Specializations and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19374.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19474.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19474.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19474.14External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

75 Topologist’s sine curve 19675.1 Image of the curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19675.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19675.3 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19775.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

76 Trivial topology 19876.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19976.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

77 Tychonoff plank 20077.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20077.2 Deleted form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20077.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20077.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20077.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

78 Tychonoff space 20178.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20178.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20178.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

78.3.1 Preservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20278.3.2 Real-valued continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20278.3.3 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20378.3.4 Compactifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20378.3.5 Uniform structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

78.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20378.5 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 204

78.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20478.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20978.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

Chapter 1

a-paracompact space

In mathematics, in the field of topology, a topological space is said to be a-paracompact if every open cover of thespace has a locally finite refinement. In contrast to the definition of paracompactness, the refinement is not requiredto be open.Every paracompact space is a-paracompact, and in regular spaces the two notions coincide.

1.1 References• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

1

Chapter 2

Adjunction space

Inmathematics, an adjunction space (or attaching space) is a common construction in topologywhere one topologicalspace is attached or “glued” onto another. Specifically, let X and Y be topological spaces with A a subspace of Y. Letf : A → X be a continuous map (called the attaching map). One forms the adjunction space X ∪f Y by taking thedisjoint union of X and Y and identifying x with f(x) for all x in A. Schematically,

X ∪f Y = (X ⨿ Y )/f(A) ∼ A.

Sometimes, the adjunction is written as X +f Y . Intuitively, we think of Y as being glued onto X via the map f.As a set, X ∪f Y consists of the disjoint union of X and (Y − A). The topology, however, is specified by the quotientconstruction. In the case where A is a closed subspace of Y one can show that the map X → X ∪f Y is a closedembedding and (Y − A) → X ∪f Y is an open embedding.

2.1 Examples• A common example of an adjunction space is given when Y is a closed n-ball (or cell) and A is the boundaryof the ball, the (n−1)-sphere. Inductively attaching cells along their spherical boundaries to this space resultsin an example of a CW complex.

• Adjunction spaces are also used to define connected sums of manifolds. Here, one first removes open ballsfrom X and Y before attaching the boundaries of the removed balls along an attaching map.

• If A is a space with one point then the adjunction is the wedge sum of X and Y.

• If X is a space with one point then the adjunction is the quotient Y/A.

2.2 Categorical description

The attaching construction is an example of a pushout in the category of topological spaces. That is to say, theadjunction space is universal with respect to following commutative diagram:

2

2.3. SEE ALSO 3

Here i is the inclusion map and φX, φY are the maps obtained by composing the quotient map with the canonicalinjections into the disjoint union of X and Y. One can form a more general pushout by replacing i with an arbitrarycontinuous map g — the construction is similar. Conversely, if f is also an inclusion the attaching construction is tosimply glue X and Y together along their common subspace.

2.3 See also• Quotient space

• Mapping cylinder

2.4 References• Stephen Willard, General Topology, (1970) Addison-Wesley Publishing Company, Reading Massachusetts.

(Provides a very brief introduction.)

• Adjunction space at PlanetMath.org.

• Ronald Brown, “Topology and Groupoids”, (2006) available from amazon sites. Discusses their homotopytype, and uses adjunction spaces as an introduction to (finite) cell complexes.

Chapter 3

Appert topology

In general topology, a branch of mathematics, the Appert topology, named for Appert (1934), is an example of atopology on the set Z+ = 1, 2, 3, … of positive integers.[1] To give Z+ a topology means to say which subsets of Z+

are open in a manner that satisfies certain axioms:[2]

1. The union of open sets is an open set.

2. The finite intersection of open sets is an open set.

3. Z+ and the empty set ∅ are open sets.

In the Appert topology, the open sets are those that do not contain 1, and those that asymptotically contain almostevery positive integer.

3.1 Construction

Let S be a subset of Z+, and let N(n,S) denote the number of elements of S which are less than or equal to n:

N(n, S) = #m ∈ S : m ≤ n.

In Appert’s topology, a set S is defined to be open if either it does not contain 1 or N(n,S)/n tends towards 1 as n tendstowards infinity:[1]

limn→∞

N(n, S)n

= 1.

The empty set is an open set in this topology because ∅ is a set that does not contain 1, and the whole set Z+ is alsoopen in this topology since

N(n,Z+

)= n ,

meaning that N(n,S)/n = 1 for all n.

3.2 Related topologies

The Appert topology is closely related to the Fort space topology that arises from giving the set of integers greaterthan one the discrete topology, and then taking the point 1 as the point at infinity in a one point compactification ofthe space.[1] The Fort space is a refinement of the Appert topology.

4

3.3. PROPERTIES 5

3.3 Properties

The closed subsets of Z+, equipped with the Appert topology, are the subsets S that either contain 1 or for which

limn→∞

N(n, S)n

= 0.

As a result, Z+ is a completely normal space (and thus also Hausdorff), for suppose that A and B are disjoint closedsets. If A ∪ B did not contain 1, then A and B would also be open and thus completely separated. On the other hand,if A contains 1 then B is open and limn→∞ N(n,B)/n=0 , so that Z+−B is an open neighborhood of A disjoint from B.[1]

A subset of Z+ is compact in the Appert topology if and only if it is finite. In particular, Z+ is not locally compact,since there is no compact neighborhood of 1. Moreover, Z+ is not countably compact.[1]

3.4 Notes[1] Steen & Seebach 1995, pp. 117–118

[2] Steen & Seebach 1995, p. 3

3.5 References• Appert, Q (1934), Propriétés des Espaces Abstraits les Plus Généraux, Actual. Sci. Ind. (146), Hermann.

• Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, ISBN 0-486-68735-X.

Chapter 4

Arens–Fort space

In mathematics, the Arens–Fort space is a special example in the theory of topological spaces, named for RichardFriederich Arens and M. K. Fort, Jr.Let X be a set of ordered pairs of non-negative integers (m, n). A subset U of X is open if and only if:

• it does not contain (0, 0), or

• it contains (0, 0), and all but a finite number of points of all but a finite number of columns, where a columnis a set (m, n) with fixed m.

In other words, an open set is only “allowed” to contain (0, 0) if only a finite number of its columns contain significantgaps. By a significant gap in a column we mean the omission of an infinite number of points.It is

• Hausdorff

• regular

• normal

It is not:

• second-countable

• first-countable

• metrizable

• compact

4.1 See also• Fort space

4.2 References• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446

6

Chapter 5

Box topology

In topology, the cartesian product of topological spaces can be given several different topologies. One of the moreobvious choices is the box topology, where a base is given by the Cartesian products of open sets in the componentspaces.[1] Another possibility is the product topology, where a base is given by the Cartesian products of open sets inthe component spaces, only finitely many of which can be not equal to the entire component space.While the box topology has a somewhat more intuitive definition than the product topology, it satisfies fewer desirableproperties. In particular, if all the component spaces are compact, the box topology on their Cartesian product will notnecessarily be compact, although the product topology on their Cartesian product will always be compact. In general,the box topology is finer than the product topology, although the two agree in the case of finite direct products (orwhen all but finitely many of the factors are trivial).

5.1 Definition

Given X such that

X :=∏i∈I

Xi,

or the (possibly infinite) Cartesian product of the topological spaces Xi , indexed by i ∈ I , the box topology on Xis generated by the base

B =

∏i∈I

Ui

∣∣∣Ui in open Xi

.

The name box comes from the case of Rn, the basis sets look like boxes or unions thereof.

5.2 Properties

Box topology on Rω:[2]

• The box topology is completely regular

• The box topology is neither compact nor connected

• The box topology is not first countable (hence not metrizable)

• The box topology is not separable

• The box topology is paracompact (and hence normal and completely regular) if the continuum hypothesis istrue

7

8 CHAPTER 5. BOX TOPOLOGY

5.2.1 Example - Failure at continuity

The following example is based on the Hilbert cube. Let Rω denote the countable cartesian product of R with itself,i.e. the set of all sequences in R. Equip R with the standard topology and Rω with the box topology. Let f : R→ Rω

be the product map whose components are all the identity, i.e. f(x) = (x, x, x, ...). Although each component functionis continuous, f is not continuous. To see this, consider the open set U = Π∞n=1(-1⁄n, 1⁄n). Since f(0) = (0, 0, 0, ...)∈U, if f were continuous, then there would exist some ε>0 such that (-ε,ε)∈f−1(U). But this would imply that f(ε⁄2)=(ε⁄2, ε ⁄2, ε⁄2, ...)∈U which is false since ε ⁄2 > 1⁄n for n > ⌈2⁄ε⌉. Thus f isnot continuous even though all its component functions are.

5.2.2 Example - Failure at compactness

Consider the countable product X =∏

Xi where for each i, Xi = 0, 1 with the discrete topology. The boxtopology on X will also be the discrete topology. Consider the sequence xn∞n=1 given by

(xn)m =

0 m < n

1 m ≥ n

Since no two points in the sequence are the same, the sequence has no limit point, and therefore X is not compact,even though its component spaces are.

5.2.3 Intuitive Description of Convergence; Comparisons

Topologies are often best understood by describing how sequences converge. In general, a cartesian product of aspace X with itself over an indexing set S is precisely the space of functions from S to X; the product topology yieldsthe topology of pointwise convergence; sequences of functions converge if and only if they converge at every pointof S. The box topology, once again due to its great profusion of open sets, makes convergence very hard. One way tovisualize the convergence in this topology is to think of functions from R to R— a sequence of functions convergesto a function f in the box topology if, when looking at the graph of f, given any set of “hoops”, that is, verticalopen intervals surrounding the graph of f above every point on the x-axis, eventually, every function in the sequence“jumps through all the hoops.” For functions on R this looks a lot like uniform convergence, in which case all the“hoops”, once chosen, must be the same size. But in this case one can make the hoops arbitrarily small, so one cansee intuitively how “hard” it is for sequences of functions to converge. The hoop picture works for convergence inthe product topology as well: here we only require all the functions to jump through any given finite set of hoops.This stems directly from the fact that, in the product topology, almost all the factors in a basic open set are the wholespace. Interestingly, this is actually equivalent to requiring all functions to eventually jump through just a single givenhoop; this is just the definition of pointwise convergence.

5.3 Comparison with product topology

The basis sets in the product topology have almost the same definition as the above, except with the qualification thatall but finitely many Ui are equal to the component space Xi. The product topology satisfies a very desirable propertyfor maps fi : Y → Xi into the component spaces: the product map f: Y → X defined by the component functions f iscontinuous if and only if all the fi are continuous. As shown above, this does not always hold in the box topology. Thisactually makes the box topology very useful for providing counterexamples — many qualities such as compactness,connectedness, metrizability, etc., if possessed by the factor spaces, are not in general preserved in the product withthis topology.

5.4 See also

• Cylinder set

5.5. NOTES 9

5.5 Notes[1] Willard, 8.2 pp. 52–53,

[2] Steen, Seebach, 109. pp. 128–129.

5.6 References• Steen, Lynn A. and Seebach, J. Arthur Jr.; Counterexamples in Topology, Holt, Rinehart and Winston (1970).ISBN 0030794854.

• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

5.7 External links• Box topology at PlanetMath.org.

Chapter 6

Cantor set

In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkableand deep properties. It was discovered in 1874 by Henry John Stephen Smith[1][2][3][4] and introduced by Germanmathematician Georg Cantor in 1883.[5][6]

Through consideration of this set, Cantor and others helped lay the foundations of modern point-set topology. Al-though Cantor himself defined the set in a general, abstract way, the most commonmodern construction is theCantorternary set, built by removing the middle thirds of a line segment. Cantor himself only mentioned the ternary con-struction in passing, as an example of a more general idea, that of a perfect set that is nowhere dense.

6.1 Construction and formula of the ternary set

The Cantor ternary set is created by repeatedly deleting the open middle third of a set of line segments. One starts bydeleting the open middle third (1⁄3, 2⁄3) from the interval [0, 1], leaving two line segments: [0, 1⁄3] ∪ [2⁄3, 1]. Next,the open middle third of each of these remaining segments is deleted, leaving four line segments: [0, 1⁄9] ∪ [2⁄9, 1⁄3]∪ [2⁄3, 7⁄9] ∪ [8⁄9, 1]. This process is continued ad infinitum, where the nth set is

Cn = Cn−1

3 ∪(

23 + Cn−1

3

)and C0 = [0, 1].

The Cantor ternary set contains all points in the interval [0, 1] that are not deleted at any step in this infinite process.The first six steps of this process are illustrated below.

An explicit closed formula for the Cantor set is

C =∞∩

m=1

3m−1−1∩k=0

([0,

3k + 1

3m

]∪[3k + 2

3m, 1

])or

C = [0, 1] \∞∪

m=1

3m−1−1∪k=0

(3k + 1

3m,3k + 2

3m

).

10

6.2. COMPOSITION 11

The proof of the formula above as the special case of two family of Cantor sets is done by the idea of self-similaritytransformations and can be found in detail.[7][8]

This process of removing middle thirds is a simple example of a finite subdivision rule.It is perhaps most intuitive to think about the Cantor set as the set of real numbers between zero and one whoseternary expansion in base three doesn't contain the digit 1. This ternary digit expansion description has been more ofinterest for researchers to explore fractal and topological properties of the Cantor set.

6.2 Composition

Since the Cantor set is defined as the set of points not excluded, the proportion (i.e., measure) of the unit intervalremaining can be found by total length removed. This total is the geometric progression

∞∑n=0

2n

3n+1=

1

3+

2

9+

4

27+

8

81+ · · · = 1

3

(1

1− 23

)= 1.

So that the proportion left is 1 – 1 = 0.This calculation shows that the Cantor set cannot contain any interval of non-zero length. In fact, it may seemsurprising that there should be anything left — after all, the sum of the lengths of the removed intervals is equalto the length of the original interval. However, a closer look at the process reveals that there must be somethingleft, since removing the “middle third” of each interval involved removing open sets (sets that do not include theirendpoints). So removing the line segment (1/3, 2/3) from the original interval [0, 1] leaves behind the points 1/3 and2/3. Subsequent steps do not remove these (or other) endpoints, since the intervals removed are always internal to theintervals remaining. So the Cantor set is not empty, and in fact contains an uncountably infinite number of points.It may appear that only the endpoints are left, but that is not the case either. The number 1/4, for example, is in thebottom third, so it is not removed at the first step, and is in the top third of the bottom third, and is in the bottomthird of that, and in the top third of that, and so on ad infinitum—alternating between top and bottom thirds. Sinceit is never in one of the middle thirds, it is never removed, and yet it is also not one of the endpoints of any middlethird. The number 3/10 is also in the Cantor set and is not an endpoint.In the sense of cardinality, most members of the Cantor set are not endpoints of deleted intervals. Since each stepremoves a finite number of intervals and the number of steps is countable, the set of endpoints is countable while thewhole Cantor set is uncountable.

6.3 Properties

6.3.1 Cardinality

It can be shown that there are as many points left behind in this process as there were to begin with, and that therefore,the Cantor set is uncountable. To see this, we show that there is a function f from the Cantor set C to the closedinterval [0,1] that is surjective (i.e. f maps from C onto [0,1]) so that the cardinality of C is no less than that of [0,1].Since C is a subset of [0,1], its cardinality is also no greater, so the two cardinalities must in fact be equal, by theCantor–Bernstein–Schroeder theorem.To construct this function, consider the points in the [0, 1] interval in terms of base 3 (or ternary) notation. Recallthat some points admit more than one representation in this notation, as for example 1/3, that can be written as0.13 but also as 0.022222...3, and 2/3, that can be written as 0.23 but also as 0.12222...3. (This alternative recurringrepresentation of a number with a terminating numeral occurs in any positional system.) When we remove the middlethird, this contains the numbers with ternary numerals of the form 0.1xxxxx...3 where xxxxx...3 is strictly between00000...3 and 22222...3. So the numbers remaining after the first step consist of

• Numbers of the form 0.0xxxxx...3

• 1/3 = 0.13 = 0.022222...3

12 CHAPTER 6. CANTOR SET

• 2/3 = 0.122222...3 = 0.23• Numbers of the form 0.2xxxxx...3.

This can be summarized by saying that those numbers that admit a ternary representation such that the first digit afterthe decimal point is not 1 are the ones remaining after the first step.The second step removes numbers of the form 0.01xxxx...3 and 0.21xxxx...3, and (with appropriate care for theendpoints) it can be concluded that the remaining numbers are those with a ternary numeral where neither of the firsttwo digits is 1. Continuing in this way, for a number not to be excluded at step n, it must have a ternary representationwhose nth digit is not 1. For a number to be in the Cantor set, it must not be excluded at any step, it must admit anumeral representation consisting entirely of 0s and 2s. It is worth emphasising that numbers like 1, 1/3 = 0.13 and7/9 = 0.213 are in the Cantor set, as they have ternary numerals consisting entirely of 0s and 2s: 1 = 0.2222...3, 1/3 =0.022222...3 and 7/9 = 0.2022222...3. So while a number in C may have either a terminating or a recurring ternarynumeral, one of its representations will consist entirely of 0s and 2s.The function from C to [0,1] is defined by taking the numeral that does consist entirely of 0s and 2s, replacing all the2s by 1s, and interpreting the sequence as a binary representation of a real number. In a formula,

f

( ∞∑k=1

ak3−k

)=

∞∑k=1

ak22−k.

For any number y in [0,1], its binary representation can be translated into a ternary representation of a numberx in C by replacing all the 1s by 2s. With this, f(x) = y so that y is in the range of f. For instance if y = 3/5= 0.100110011001...2, we write x = 0.200220022002...3 = 7/10. Consequently f is surjective; however, f is notinjective — interestingly enough, the values for which f(x) coincides are those at opposing ends of one of the middlethirds removed. For instance, 7/9 = 0.2022222...3 and 8/9 = 0.2200000...3 so f(7/9) = 0.101111...2 = 0.112 = f(8/9).So there are as many points in the Cantor set as there are in [0, 1], and the Cantor set is uncountable (see Cantor’sdiagonal argument). However, the set of endpoints of the removed intervals is countable, so theremust be uncountablymany numbers in the Cantor set which are not interval endpoints. As noted above, one example of such a number is¼, which can be written as 0.02020202020...3 in ternary notation.The Cantor set contains as many points as the interval from which it is taken, yet itself contains no interval of nonzerolength. The irrational numbers have the same property, but the Cantor set has the additional property of being closed,so it is not even dense in any interval, unlike the irrational numbers which are dense in every interval.It has been conjectured that all algebraic irrational numbers are normal. Since members of the Cantor set are notnormal, this would imply that all members of the Cantor set are either rational or transcendental.

6.3.2 Self-similarity

The Cantor set is the prototype of a fractal. It is self-similar, because it is equal to two copies of itself, if each copyis shrunk by a factor of 3 and translated. More precisely, there are two functions, the left and right self-similaritytransformations, fL(x) = x/3 and fR(x) = (2+x)/3 , which leave the Cantor set invariant up to homeomorphism:fL(C) ∼= fR(C) ∼= C.

Repeated iteration of fL and fR can be visualized as an infinite binary tree. That is, at each node of the tree, onemay consider the subtree to the left or to the right. Taking the set fL, fR together with function composition formsa monoid, the dyadic monoid.The automorphisms of the binary tree are its hyperbolic rotations, and are given by the modular group. Thus, theCantor set is a homogeneous space in the sense that for any two points x and y in the Cantor set C , there existsa homeomorphism h : C → C with h(x) = y . These homeomorphisms can be expressed explicitly, as Möbiustransformations.The Hausdorff dimension of the Cantor set is equal to ln(2)/ln(3) ≈ 0.631.

6.3.3 Topological and analytical properties

Although “the” Cantor set typically refers to the original, middle-thirds Cantor desecribed above, topologists oftentalk about “a” Cantor set, which means any topological space that is homeomorphic (topologically equivalent) to it.

6.3. PROPERTIES 13

As the above summation argument shows, the Cantor set is uncountable but has Lebesgue measure 0. Since theCantor set is the complement of a union of open sets, it itself is a closed subset of the reals, and therefore a completemetric space. Since it is also totally bounded, the Heine–Borel theorem says that it must be compact.For any point in the Cantor set and any arbitrarily small neighborhood of the point, there is some other number witha ternary numeral of only 0s and 2s, as well as numbers whose ternary numerals contain 1s. Hence, every point inthe Cantor set is an accumulation point (also called a cluster point or limit point) of the Cantor set, but none is aninterior point. A closed set in which every point is an accumulation point is also called a perfect set in topology, whilea closed subset of the interval with no interior points is nowhere dense in the interval.Every point of the Cantor set is also an accumulation point of the complement of the Cantor set.For any two points in the Cantor set, there will be some ternary digit where they differ — one will have 0 and theother 2. By splitting the Cantor set into “halves” depending on the value of this digit, one obtains a partition of theCantor set into two closed sets that separate the original two points. In the relative topology on the Cantor set, thepoints have been separated by a clopen set. Consequently the Cantor set is totally disconnected. As a compact totallydisconnected Hausdorff space, the Cantor set is an example of a Stone space.As a topological space, the Cantor set is naturally homeomorphic to the product of countably many copies of thespace 0, 1 , where each copy carries the discrete topology. This is the space of all sequences in two digits

2N = (xn)|xn ∈ 0, 1 for n ∈ N

which can also be identified with the set of 2-adic integers. The basis for the open sets of the product topology arecylinder sets; the homeomorphism maps these to the subspace topology that the Cantor set inherits from the naturaltopology on the real number line. This characterization of the Cantor space as a product of compact spaces gives asecond proof that Cantor space is compact, via Tychonoff’s theorem.From the above characterization, the Cantor set is homeomorphic to the p-adic integers, and, if one point is removedfrom it, to the p-adic numbers.The Cantor set is a subset of the reals, which are a metric space with respect to the ordinary distance metric; thereforethe Cantor set itself is a metric space, by using that same metric. Alternatively, one can use the p-adic metric on 2N: given two sequences (xn), (yn) ∈ 2N , the distance between them is d((xn), (yn)) = 1/k , where k is the smallestindex such that xk = yk ; if there is no such index, then the two sequences are the same, and one defines the distanceto be zero. These two metrics generate the same topology on the Cantor set.We have seen above that the Cantor set is a totally disconnected perfect compact metric space. Indeed, in a sense itis the only one: every nonempty totally disconnected perfect compact metric space is homeomorphic to the Cantorset. See Cantor space for more on spaces homeomorphic to the Cantor set.The Cantor set is sometimes regarded as “universal” in the category of compact metric spaces, since any compactmetric space is a continuous image of the Cantor set; however this construction is not unique and so the Cantor setis not universal in the precise categorical sense. The “universal” property has important applications in functionalanalysis, where it is sometimes known as the representation theorem for compact metric spaces.[9]

For any integer q≥ 2, the topology on the groupG=Zqω (the countable direct sum) is discrete. Although the Pontrjagindual Γ is also Zqω, the topology of Γ is compact. One can see that Γ is totally disconnected and perfect - thus it ishomeomorphic to the Cantor set. It is easiest to write out the homeomorphism explicitly in the case q=2. (See Rudin1962 p 40.)

6.3.4 Measure and probability

The Cantor set can be seen as the compact group of binary sequences, and as such, it is endowed with a natural Haarmeasure. When normalized so that the measure of the set is 1, it is a model of an infinite sequence of coin tosses.Furthermore, one can show that the usual Lebesgue measure on the interval is an image of the Haar measure on theCantor set, while the natural injection into the ternary set is a canonical example of a singular measure. It can alsobe shown that the Haar measure is an image of any probability, making the Cantor set a universal probability spacein some ways.In Lebesgue measure theory, the Cantor set is an example of a set which is uncountable and has zero measure.[10]

14 CHAPTER 6. CANTOR SET

6.4 Variants

6.4.1 Smith–Volterra–Cantor set

Main article: Smith–Volterra–Cantor set

Instead of repeatedly removing the middle third of every piece as in the Cantor set, we could also keep removing anyother fixed percentage (other than 0% and 100%) from the middle. In the case where the middle 8/10 of the intervalis removed, we get a remarkably accessible case — the set consists of all numbers in [0,1] that can be written as adecimal consisting entirely of 0s and 9s.By removing progressively smaller percentages of the remaining pieces in every step, one can also construct setshomeomorphic to the Cantor set that have positive Lebesgue measure, while still being nowhere dense. See Smith–Volterra–Cantor set for an example.

6.4.2 Cantor dust

Cantor dust is a multi-dimensional version of the Cantor set. It can be formed by taking a finite Cartesian productof the Cantor set with itself, making it a Cantor space. Like the Cantor set, Cantor dust has zero measure.[11]

A different 2D analogue of the Cantor set is the Sierpinski carpet, where a square is divided up into nine smallersquares, and the middle one removed. The remaining squares are then further divided into nine each and the middleremoved, and so on ad infinitum.[12] The 3D analogue of this is the Menger sponge.

6.5 Historical remarks

Cantor himself defined the set in a general, abstract way, and mentioned the ternary construction only in passing, asan example of a more general idea, that of a perfect set that is nowhere dense. The original paper provides severaldifferent constructions of the abstract concept.This set would have been considered abstract at the time when Cantor devised it. Cantor himself was led to it bypractical concerns about the set of points where a trigonometric series might fail to converge. The discovery did muchto set him on the course for developing an abstract, general theory of infinite sets.

6.6 See also• Cantor function• Cantor cube• Antoine’s necklace• Koch snowflake• Knaster–Kuratowski fan• List of fractals by Hausdorff dimension

6.7 Notes[1] Henry J.S. Smith (1874) “On the integration of discontinuous functions.” Proceedings of the London Mathematical Society,

Series 1, vol. 6, pages 140–153.

[2] The “Cantor set” was also discovered by Paul du Bois-Reymond (1831–1889). See footnote on page 128 of: Paul duBois-Reymond (1880) “Der Beweis des Fundamentalsatzes der Integralrechnung,”Mathematische Annalen, vol. 16, pages115–128. The “Cantor set” was also discovered in 1881 by Vito Volterra (1860–1940). See: Vito Volterra (1881) “Alcuneosservazioni sulle funzioni punteggiate discontinue” [Some observations on point-wise discontinuous functions], Giornaledi Matematiche, vol. 19, pages 76–86.

6.8. REFERENCES 15

[3] José Ferreirós, Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics (Basel, Switzerland:Birkhäuser Verlag, 1999), pages 162–165.

[4] Ian Stewart, Does God Play Dice?: The New Mathematics of Chaos

[5] Georg Cantor (1883) "Über unendliche, lineare Punktmannigfaltigkeiten V" [On infinite, linear point-manifolds (sets)],Mathematische Annalen, vol. 21, pages 545–591.

[6] H.-O. Peitgen, H. Jürgens, and D. Saupe, Chaos and Fractals: New Frontiers of Science 2nd ed. (N.Y., N.Y.: SpringerVerlag, 2004), page 65.

[7] Mohsen Soltanifar, On A sequence of cantor Fractals, Rose Hulman Undergraduate Mathematics Journal, Vol 7, No 1,paper 9, 2006.

[8] Mohsen Soltanifar, A Different Description of A Family of Middle-a Cantor Sets, American Journal of UndergraduateResearch, Vol 5, No 2, pp 9–12, 2006.

[9] Stephen Willard, General Topology, Addison-Wesley Publishing Company, 1968.

[10] the Cantor set is an uncountable set with zero measure

[11] Helmberg, Gilbert (2007). Getting Acquainted With Fractals. Walter de Gruyter. p. 46. ISBN 978-3-11-019092-2.

[12] Helmberg, Gilbert (2007). Getting Acquainted With Fractals. Walter de Gruyter. p. 48. ISBN 978-3-11-019092-2.

6.8 References• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446 (See example 29).

• Gary L. Wise and Eric B. Hall, Counterexamples in Probability and Real Analysis. Oxford University Press,New York 1993. ISBN 0-19-507068-2. (See chapter 1).

6.9 External links• Hazewinkel, Michiel, ed. (2001), “Cantor set”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Cantor Sets and Cantor Set and Function at cut-the-knot

• Cantor Set (PRIME)

• Cantor Dust Demo Program

Chapter 7

Cantor space

In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: atopological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ωis called “the” Cantor space. Note that, commonly, 2ω is referred to simply as the Cantor set, while the term Cantorspace is reserved for the more general construction ofDS for a finite setD and a set S which might be finite, countableor possibly uncountable.[1]

7.1 Examples

The Cantor set itself is a Cantor space. But the canonical example of a Cantor space is the countably infinitetopological product of the discrete 2-point space 0, 1. This is usually written as 2N or 2ω (where 2 denotes the2-element set 0,1 with the discrete topology). A point in 2ω is an infinite binary sequence, that is a sequence whichassumes only the values 0 or 1. Given such a sequence a0, a1, a2,..., one can map it to the real number

∞∑n=0

2an3n+1

.

This mapping gives a homeomorphism from 2ω onto the Cantor set, demonstrating that 2ω is indeed a Cantor space.Cantor spaces occur abundantly in real analysis. For example, they exist as subspaces in every perfect, completemetric space. (To see this, note that in such a space, any non-empty perfect set contains two disjoint non-emptyperfect subsets of arbitrarily small diameter, and so one can imitate the construction of the usual Cantor set.) Also,every uncountable, separable, completely metrizable space contains Cantor spaces as subspaces. This includes mostof the common type of spaces in real analysis.

7.2 Characterization

A topological characterization of Cantor spaces is given by Brouwer's theorem:[2]

Any two non-empty compact Hausdorff spaces without isolated points and having countable bases consist-ing of clopen sets are homeomorphic to each other.

The topological property of having a base consisting of clopen sets is sometimes known as “zero-dimensionality”.Brouwer’s theorem can be restated as:

A topological space is a Cantor space if and only if it is non-empty, perfect, compact, totally disconnected,and metrizable.

This theorem is also equivalent (via Stone’s representation theorem for Boolean algebras) to the fact that any twocountable atomless Boolean algebras are isomorphic.

16

7.3. PROPERTIES 17

7.3 Properties

As can be expected from Brouwer’s theorem, Cantor spaces appear in several forms. But many properties of Cantorspaces can be established using 2ω, because its construction as a product makes it amenable to analysis.Cantor spaces have the following properties:

• The cardinality of any Cantor space is 2ℵ0 , that is, the cardinality of the continuum.

• The product of two (or even any finite or countable number of) Cantor spaces is a Cantor space. Along withthe Cantor function; this fact can be used to construct space-filling curves.

• A Hausdorff topological space is compact metrizable if and only if it is a continuous image of a Cantorspace.[3][4]

Let C(X) denote the space of all real-valued, bounded continuous functions on a topological space X. Let K denote acompact metric space, and Δ denote the Cantor set. Then the Cantor set has the following property:

• C(K) is isometric to a closed subspace of C(Δ).[5]

In general, this isometry is not unique, and thus is not properly a universal property in the categorical sense.

• The group of all homeomorphisms of the Cantor space is simple.[6]

7.4 See also• Space (mathematics)

• Cantor set

• Cantor cube

7.5 References[1] Stephen Willard, General Topology (1970) Addison-Wesley Publishing. See section 17.9a

[2] Brouwer, L. E. J. (1910), “On the structure of perfect sets of points” (PDF), Proc. Koninklijke Akademie vanWetenschappen12: 785–794.

[3] N.L. Carothers, A Short Course on Banach Space Theory, London Mathematical Society Student Texts 64, (2005) Cam-bridge University Press. See Chapter 12

[4] Willard, op.cit., See section 30.7

[5] Carothers, op.cit.

[6] R.D. Anderson, The Algebraic Simplicity of Certain Groups of Homeomorphisms, American Journal of Mathematics 80(1958), pp. 955-963.

• Kechris, A. (1995). Classical Descriptive Set Theory (Graduate Texts inMathematics 156 ed.). Springer. ISBN0-387-94374-9.

Chapter 8

Comb space

In mathematics, particularly topology, a comb space is a subspace of R2 that looks rather like a comb. The combspace has some rather interesting properties and provides interesting counterexamples. The topologist’s sine curvehas similar properties to the comb space. The deleted comb space is an important variation on the comb space.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

Topologist's comb

Topologist’s comb

8.1 Formal definition

Consider R2 with its standard topology and let K be the set 1/n|n ∈ N \ 0 . The set C defined by:

(0 × [0, 1]) ∪ (K × [0, 1]) ∪ ([0, 1]× 0)

18

8.2. TOPOLOGICAL PROPERTIES 19

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

The intricated double comb for r=3/4

The intricated double comb for r=3/4.

considered as a subspace of R2 equipped with the subspace topology is known as the comb space. The deleted combspace, D, is defined by:

(0 × 0, 1) ∪ (K × [0, 1]) ∪ ([0, 1]× 0)

This is the comb space with the line segment 0 × (0, 1) deleted.

8.2 Topological properties

The comb space and the deleted comb space have some interesting topological properties mostly related to the notionof connectedness.1. The comb space is an example of a path connected space which is not locally path connected.2. The deleted comb space, D, is connected:

Let E be the comb space without 0 × (0, 1] . E is also path connected and the closure ofE is the comb space. As E ⊂ D ⊂ the closure of E, where E is connected, the deleted combspace is also connected.

3. The deleted comb space is not path connected since there is no path from (0,1) to (0,0):

Suppose there is a path from p = (0, 1) to a point q in D. Let ƒ : [0, 1] → D be this path.We shall prove that ƒ −1p is both open and closed in [0, 1] contradicting the connectedness

20 CHAPTER 8. COMB SPACE

of this set. Clearly we have ƒ −1p is closed in [0, 1] by the continuity of ƒ. To prove thatƒ −1p is open, we proceed as follows: Choose a neighbourhood V (open in R2) about pthat doesn’t intersect the x–axis. Suppose x is an arbitrary point in ƒ −1p. Clearly, f(x) =p. Then since f −1(V) is open, there is a basis element U containing x such that ƒ(U) is asubset of V. We assert that ƒ(U) = p which will mean that U is an open subset of ƒ −1pcontaining x. Since x was arbitrary, ƒ −1p will then be open. We know that U is connectedsince it is a basis element for the order topology on [0, 1]. Therefore, ƒ(U) is connected.Suppose ƒ(U) contains a point s other than p. Then s = (1/n, z) must belong to D. Choose rsuch that 1/(n + 1) < r < 1/n. Since ƒ(U) does not intersect the x-axis, the sets A = (−∞, r)× R and B = (r, +∞) × R will form a separation on f(U); contradicting the connectedness off(U). Therefore, f −1p is both open and closed in [0, 1]. This is a contradiction.

8.3 See also• Locally connected space

• Connected space

• Topologist’s sine curve

• Infinite broom

• Order topology

8.4 References• James Munkres (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.

• Kiyosi Ito (ed.). “Connectedness”. Encyclopedic Dictionary of Mathematics. Mathematical Society of Japan.

Chapter 9

Compact convergence

In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence whichgeneralizes the idea of uniform convergence. It is associated with the compact-open topology.

9.1 Definition

Let (X, T ) be a topological space and (Y, dY ) be a metric space. A sequence of functions

fn : X → Y , n ∈ N,

is said to converge compactly as n → ∞ to some function f : X → Y if, for every compact setK ⊆ X ,

(fn)|K → f |K

converges uniformly onK as n → ∞ . This means that for all compactK ⊆ X ,

limn→∞

supx∈K

dY (fn(x), f(x)) = 0.

9.2 Examples• If X = (0, 1) ⊂ R and Y = R with their usual topologies, with fn(x) := xn , then fn converges compactlyto the constant function with value 0, but not uniformly.

• If X = (0, 1] , Y = R and fn(x) = xn , then fn converges pointwise to the function that is zero on (0, 1)and one at 1 , but the sequence does not converge compactly.

• A very powerful tool for showing compact convergence is the Arzelà–Ascoli theorem. There are several ver-sions of this theorem, roughly speaking it states that every sequence of equicontinuous and uniformly boundedmaps has a subsequence which converges compactly to some continuous map.

9.3 Properties• If fn → f uniformly, then fn → f compactly.

• If (X, T ) is a compact space and fn → f compactly, then fn → f uniformly.

• If (X, T ) is locally compact, then fn → f compactly if and only if fn → f locally uniformly.

• If (X, T ) is a compactly generated space, fn → f compactly, and each fn is continuous, then f is continuous.

21

22 CHAPTER 9. COMPACT CONVERGENCE

9.4 See also• Modes of convergence (annotated index)

• Montel’s theorem

9.5 References• R. Remmert Theory of complex functions (1991 Springer) p. 95

Chapter 10

Cosmic space

In mathematics, particularly topology, a cosmic space is any topological space that is a continuous image of someseparable metric space. Equivalently (for regular T1 spaces but not in general), a space is cosmic if and only if it hasa countable network; namely a countable collection of subsets of the space such that any open set is the union of asubcollection of these sets.Cosmic spaces have several interesting properties. There are a number of unsolved problems about them.

10.1 Examples and properties• Any open subset of a cosmic space is cosmic since open subsets of separable spaces are separable.

• Separable metric spaces are trivially cosmic.

10.2 Unsolved problems

It is unknown as to whether X is cosmic if:a) X2 contains no uncountable discrete space;b) the countable product of X with itself is hereditarily separable and hereditarily Lindelöf.

10.3 References• Deza,MichelMarie; Deza, Elena (2012). Encyclopedia ofDistances. Springer-Verlag. p. 64. ISBN3642309585.

• Hart, K.P.; Nagata, Jun-iti; Vaughan, J.E. (2003). Encyclopedia of General Topology. Elsevier. p. 273. ISBN0080530869.

10.4 External links• A book of unsolved problems in topology; see page 91 for cosmic spaces

23

Chapter 11

CW complex

In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs ofhomotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes,but still retains a combinatorial nature that allows for computation (often with a much smaller complex).

11.1 Formulation

Roughly speaking, a CW complex is made of basic building blocks called cells. The precise definition prescribes howthe cells may be topologically glued together. The C stands for “closure-finite”, and theW for "weak topology".An n-dimensional closed cell is the image of an n-dimensional closed ball under an attaching map. For example,a simplex is a closed cell, and more generally, a convex polytope is a closed cell. An n-dimensional open cell is atopological space that is homeomorphic to the n-dimensional open ball. A 0-dimensional open (and closed) cell is asingleton space. Closure-finite means that each closed cell is covered by a finite union of open cells.A CW complex is a Hausdorff space X together with a partition of X into open cells (of perhaps varying dimension)that satisfies two additional properties:

• For each n-dimensional open cellC in the partition ofX, there exists a continuousmap f from the n-dimensionalclosed ball to X such that

• the restriction of f to the interior of the closed ball is a homeomorphism onto the cell C, and• the image of the boundary of the closed ball is contained in the union of a finite number of elements ofthe partition, each having cell dimension less than n.

• A subset of X is closed if and only if it meets the closure of each cell in a closed set.

11.2 Inductive definition of CW complexes

If the largest dimension of any of the cells is n, then the CW complex is said to have dimension n. If there is no boundto the cell dimensions then it is said to be infinite-dimensional. The n-skeleton of a CW complex is the union of thecells whose dimension is at most n. If the union of a set of cells is closed, then this union is itself a CW complex,called a subcomplex. Thus the n-skeleton is the largest subcomplex of dimension n or less.A CW complex is often constructed by defining its skeleta inductively. Begin by taking the 0-skeleton to be a discretespace. Next, attach 1-cells to the 0-skeleton. Here, each 1-cell begins as a closed 1-ball and is attached to the 0-skeleton via some (continuous) map from the boundary of the 1-ball, that is, from the 0-sphere S0 . Each point of S0

can be identified with its image in the 0-skeleton under the aforementioned map; this is an equivalence relation. The1-skeleton is then defined to be the identification space obtained from the union of the 0-skeleton and 1-cells underthis equivalence relation.In general, given the (n − 1)-skeleton, the n-skeleton is formed by attaching n-cells to it. Each n-cell begins as aclosed n-ball and is attached to the (n − 1)-skeleton via some continuous map from the boundary of the n-ball, that

24

11.3. EXAMPLES 25

is, from the (n − 1)-sphere Sn−1 . Each point of Sn−1 can be identified with its image in the (n − 1)-skeleton underthe aforementioned map; this is again an equivalence relation. The n-skeleton is then defined to be the identificationspace obtained from the union of the (n − 1)-skeleton and n-cells under this equivalence relation.Up to isomorphism every n-dimensional complex can be obtained from its (n − 1)-skeleton in this sense, and thusevery finite-dimensional CW complex can be built up by the process above. This is true even for infinite-dimensionalcomplexes, with the understanding that the result of the infinite process is the direct limit of the skeleta: a set is closedin X if and only if it meets each skeleton in a closed set.

11.3 Examples• The standard CW structure on the real numbers has 0-skeleton the integersZ and as 1-cells the intervals [n, n+

1] : n ∈ Z . Similarly, the standard CW structure onRn has cubical cells that are products of the 0 and 1-cellsfrom R . This is the standard cubic lattice cell structure on Rn .

• A polyhedron is naturally a CW complex.

• A graph is a 1-dimensional CW complex. Trivalent graphs can be considered as generic 1-dimensional CWcomplexes. Specifically, if X is a 1-dimensional CW complex, the attaching map for a 1-cell is a map from atwo-point space to X, f : 0, 1 → X . This map can be perturbed to be disjoint from the 0-skeleton of X ifand only if f(0) and f(1) are not 0-valence vertices of X.

• An infinite-dimensional Hilbert space is not a CW complex: it is a Baire space and therefore cannot be writtenas a countable union of n-skeletons, each of which being a closed set with empty interior. This argumentextends to many other infinite-dimensional spaces.

• The terminology for a generic 2-dimensional CW complex is a shadow.[1]

• The n-dimensional sphere admits a CW structure with two cells, one 0-cell and one n-cell. Here the n-cellis attached by the constant mapping from Sn−1 to 0-cell. There is a popular alternative cell decomposition,since the equatorial inclusion Sn−1 → Sn has complement two balls: the upper and lower hemi-spheres.Inductively, this gives Sn a CW decomposition with two cells in every dimension k such that 0 ≤ k ≤ n .

• The n-dimensional real projective space admits a CW structure with one cell in each dimension.

• Grassmannian manifolds admit a CW structure called Schubert cells.

• Differentiable manifolds, algebraic and projective varieties have the homotopy-type of CW complexes.

• The one-point compactification of a cusped hyperbolic manifold has a canonical CW decomposition with onlyone 0-cell (the compactification point) called the Epstein-Penner Decomposition. Such cell decompositionsare frequently called ideal polyhedral decompositions and are used in popular computer software, such asSnapPea.

• The space re2πiθ : 0 ≤ r ≤ 1, θ ∈ Q ⊂ C has the homotopy-type of a CW complex (it is contractible) butit does not admit a CW decomposition, since it is not locally contractible.

• The Hawaiian earring is an example of a topological space that does not have the homotopy-type of a CWcomplex.

11.4 Homology and cohomology of CW complexes

Singular homology and cohomology of CW complexes is readily computable via cellular homology. Moreover, inthe category of CW complexes and cellular maps, cellular homology can be interpreted as a homology theory. Tocompute an extraordinary (co)homology theory for a CW complex, the Atiyah-Hirzebruch spectral sequence is theanalogue of cellular homology.Some examples:

• For the sphere Sn , take the cell decomposition with two cells: a single 0-cell and a single n-cell.The cellular homology chain complex C∗ and homology are given by:

26 CHAPTER 11. CW COMPLEX

Ck =

Z k ∈ 0, n0 k /∈ 0, n Hk =

Z k ∈ 0, n0 k /∈ 0, n since all the differentials are zero.

Alternatively, if we use the equatorial decomposition with two cells in every dimension Ck =

Z2 0 ≤ k ≤ n0 otherwise

and the differentials are matrices of the form(1 −11 −1

). This gives the same homology computation above, as the

chain complex is exact at all terms except C0 and Cn .

• For PnC we get similarly

Hk(PnC) =

Z for0 ≤ k ≤ 2n, even0 otherwise.

Both of the above examples are particularly simple because the homology is determined by the number of cells—i.e.:the cellular attaching maps have no role in these computations. This is a very special phenomenon and is not indicativeof the general case.

11.5 Modification of CW structures

There is a technique, developed byWhitehead, for replacing a CW complex with a homotopy-equivalent CW complexwhich has a simpler CW decomposition.Consider, for example, an arbitrary CW complex. Its 1-skeleton can be fairly complicated, being an arbitrary graph.Now consider a maximal forest F in this graph. Since it is a collection of trees, and trees are contractible, considerthe space X/ ∼ where the equivalence relation is generated by x ∼ y if they are contained in a common tree in themaximal forest F. The quotient map X → X/ ∼ is a homotopy equivalence. Moreover, X/ ∼ naturally inherits aCW structure, with cells corresponding to the cells of X which are not contained in F. In particular, the 1-skeletonof X/ ∼ is a disjoint union of wedges of circles.Another way of stating the above is that a connected CW complex can be replaced by a homotopy-equivalent CWcomplex whose 0-skeleton consists of a single point.Consider climbing up the connectivity ladder—assume X is a simply-connected CW complex whose 0-skeleton con-sists of a point. Can we, through suitable modifications, replace X by a homotopy-equivalent CW complex whereX1

consists of a single point? The answer is yes. The first step is to observe thatX1 and the attaching maps to constructX2 from X1 form a group presentation. The Tietze theorem for group presentations states that there is a sequenceof moves we can perform to reduce this group presentation to the trivial presentation of the trivial group. There aretwo Tietze moves:

1) Adding/removing a generator. Adding a generator, from the perspective of the CW decompositionconsists of adding a 1-cell and a 2-cell whose attaching map consists of the new 1-cell and the remainderof the attaching map is in X1 . If we let X be the corresponding CW complex X = X ∪ e1 ∪ e2 thenthere is a homotopy-equivalence X → X given by sliding the new 2-cell into X.

2) Adding/removing a relation. The act of adding a relation is similar, only one is replacing X by X =X ∪ e2 ∪ e3 where the new 3-cell has an attaching map that consists of the new 2-cell and remaindermapping into X2 . A similar slide gives a homotopy-equivalence X → X .

If a CW complex X is n-connected one can find a homotopy-equivalent CW complex X whose n-skeleton Xn

consists of a single point. The argument for n ≥ 2 is similar to the n = 1 case, only one replaces Tietze movesfor the fundamental group presentation by elementary matrix operations for the presentation matrices forHn(X;Z)(using the presentation matrices coming from cellular homology. i.e.: one can similarly realize elementary matrixoperations by a sequence of addition/removal of cells or suitable homotopies of the attaching maps.

11.6. 'THE' HOMOTOPY CATEGORY 27

11.6 'The' homotopy category

The homotopy category of CW complexes is, in the opinion of some experts, the best if not the only candidate for thehomotopy category (for technical reasons the version for pointed spaces is actually used).[2] Auxiliary constructionsthat yield spaces that are not CW complexes must be used on occasion. One basic result is that the representablefunctors on the homotopy category have a simple characterisation (the Brown representability theorem).

11.7 Properties

• CW complexes are locally contractible.

• CW complexes satisfy the Whitehead theorem: a map between CW complexes is a homotopy-equivalence ifand only if it induces an isomorphism on all homotopy groups.

• The product of two CW complexes can be made into a CW complex. Specifically, if X and Y are CW com-plexes, then one can form a CW complex X×Y in which each cell is a product of a cell in X and a cell in Y,endowed with the weak topology. The underlying set of X×Y is then the Cartesian product of X and Y, asexpected. In addition, the weak topology on this set often agrees with the more familiar product topology onX×Y, for example if either X or Y is finite. However, the weak topology can be finer than the product topologyif neither X nor Y is locally compact. In this unfavorable case, the product X×Y in the product topology isnot a CW complex. On the other hand, the product of X and Y in the category of compactly generated spacesagrees with the weak topology and therefore defines a CW complex.

• Let X and Y be CW complexes. Then the function spaces Hom(X,Y) (with the compact-open topology) are notCW complexes in general. If X is finite then Hom(X,Y) is homotopy equivalent to a CW complex by a theoremof John Milnor (1959).[3] Note that X and Y are compactly generated Hausdorff spaces, so Hom(X,Y) is oftentaken with the compactly generated variant of the compact-open topology; the above statements remain true.[4]

• A covering space of a CW complex is also a CW complex.

• CW complexes are paracompact. Finite CW complexes are compact. A compact subspace of a CW complexis always contained in a finite subcomplex.[5] [6]

11.8 See also

• The notion of CW complex has an adaptation to smooth manifolds called a handle decomposition which isclosely related to surgery theory.

11.9 References

11.9.1 Notes

[1] Turaev, V. G. (1994), “Quantum invariants of knots and 3-manifolds”, De Gruyter Studies in Mathematics (Berlin: Walterde Gruyter & Co.) 18

[2] For example, the opinion “The class of CW complexes (or the class of spaces of the same homotopy type as a CW complex)is the most suitable class of topological spaces in relation to homotopy theory” appears in Baladze, D.O. (2001), “CW-complex”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

[3] Milnor, John, "On spaces having the homotopy type of a CW-complex" Trans. Amer. Math. Soc. 90 (1959), 272–280.

[4] “Compactly Generated Spaces” (PDF).

[5] Hatcher, Allen, Algebraic topology, Cambridge University Press (2002). ISBN 0-521-79540-0. A free electronic versionis available on the author’s homepage

[6] Hatcher, Allen, Vector bundles and K-theory, preliminary version available on the authors homepage

28 CHAPTER 11. CW COMPLEX

11.9.2 General references

• Whitehead, J. H. C. (1949a). “Combinatorial homotopy. I.”. Bull. Amer. Math. Soc. 55 (5): 213–245.doi:10.1090/S0002-9904-1949-09175-9. MR 0030759. (open access)

• Whitehead, J. H. C. (1949b). “Combinatorial homotopy. II.”. Bull. Amer. Math. Soc. 55 (3): 453–496.doi:10.1090/S0002-9904-1949-09213-3. MR 0030760. (open access)

• Hatcher, Allen (2002). Algebraic topology. Cambridge University Press. ISBN 0-521-79540-0. This textbookdefines CW complexes in the first chapter and uses them throughout; includes an appendix on the topology ofCW complexes. A free electronic version is available on the author’s homepage.

• Lundell, A. T.; Weingram, S. (1970). The topology of CW complexes. Van Nostrand University Series inHigher Mathematics. ISBN 0-442-04910-2.

Chapter 12

Discrete space

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in whichthe points form a discontinuous sequence, meaning they are isolated from each other in a certain sense. The discretetopology is the finest topology that can be given on a set, i.e., it defines all subsets as open sets. In particular, eachsingleton is an open set in the discrete topology.

12.1 Definitions

Given a set X:

• the discrete topology on X is defined by letting every subset of X be open (and hence also closed), and X is adiscrete topological space if it is equipped with its discrete topology;

• the discrete uniformity on X is defined by letting every superset of the diagonal (x,x) : x is in X in X × Xbe an entourage, and X is a discrete uniform space if it is equipped with its discrete uniformity.

• the discrete metric ρ on X is defined by

ρ(x, y) =

1 if x = y,0 if x = y

for any x, y ∈ X . In this case (X, ρ) is called a discrete metric space or a space of isolated points.

• a set S is discrete in a metric space (X, d) , for S ⊆ X , if for every x ∈ S , there exists some δ > 0(depending on x ) such that d(x, y) > δ for all y ∈ S \ x ; such a set consists of isolated points. A set S isuniformly discrete in the metric space (X, d) , for S ⊆ X , if there exists ε > 0 such that for any two distinctx, y ∈ S , d(x, y) > ε.

A metric space (E, d) is said to be uniformly discrete if there exists a “packing radius” r > 0 such that, for anyx, y ∈ E , one has either x = y or d(x, y) > r .[1] The topology underlying a metric space can be discrete, withoutthe metric being uniformly discrete: for example the usual metric on the set 1, 1/2, 1/4, 1/8, ... of real numbers.

12.2 Properties

The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on adiscrete uniform space is the discrete topology. Thus, the different notions of discrete space are compatible with oneanother. On the other hand, the underlying topology of a non-discrete uniform or metric space can be discrete; anexample is the metric space X := 1/n : n = 1,2,3,... (with metric inherited from the real line and given by d(x,y) =|x − y|). Obviously, this is not the discrete metric; also, this space is not complete and hence not discrete as a uniformspace. Nevertheless, it is discrete as a topological space. We say that X is topologically discrete but not uniformlydiscrete or metrically discrete.Additionally:

29

30 CHAPTER 12. DISCRETE SPACE

• The topological dimension of a discrete space is equal to 0.

• A topological space is discrete if and only if its singletons are open, which is the case if and only if it doesn'tcontain any accumulation points.

• The singletons form a basis for the discrete topology.

• A uniform space X is discrete if and only if the diagonal (x,x) : x is in X is an entourage.

• Every discrete topological space satisfies each of the separation axioms; in particular, every discrete space isHausdorff, that is, separated.

• A discrete space is compact if and only if it is finite.

• Every discrete uniform or metric space is complete.

• Combining the above two facts, every discrete uniform or metric space is totally bounded if and only if it isfinite.

• Every discrete metric space is bounded.

• Every discrete space is first-countable; it is moreover second-countable if and only if it is countable.

• Every discrete space with at least two points is totally disconnected.

• Every non-empty discrete space is second category.

• Any two discrete spaces with the same cardinality are homeomorphic.

• Every discrete space is metrizable (by the discrete metric).

• A finite space is metrizable only if it is discrete.

• If X is a topological space and Y is a set carrying the discrete topology, then X is evenly covered by X × Y (theprojection map is the desired covering)

• The subspace topology on the integers as a subspace of the real line is the discrete topology.

• A discrete space is separable if and only if it is countable.

Any function from a discrete topological space to another topological space is continuous, and any function from adiscrete uniform space to another uniform space is uniformly continuous. That is, the discrete space X is free on theset X in the category of topological spaces and continuous maps or in the category of uniform spaces and uniformlycontinuous maps. These facts are examples of a much broader phenomenon, in which discrete structures are usuallyfree on sets.With metric spaces, things are more complicated, because there are several categories of metric spaces, depending onwhat is chosen for the morphisms. Certainly the discrete metric space is free when the morphisms are all uniformlycontinuous maps or all continuous maps, but this says nothing interesting about the metric structure, only the uniformor topological structure. Categories more relevant to the metric structure can be found by limiting the morphismsto Lipschitz continuous maps or to short maps; however, these categories don't have free objects (on more thanone element). However, the discrete metric space is free in the category of bounded metric spaces and Lipschitzcontinuous maps, and it is free in the category of metric spaces bounded by 1 and short maps. That is, any functionfrom a discrete metric space to another boundedmetric space is Lipschitz continuous, and any function from a discretemetric space to another metric space bounded by 1 is short.Going the other direction, a function f from a topological space Y to a discrete space X is continuous if and only ifit is locally constant in the sense that every point in Y has a neighborhood on which f is constant.

12.3. USES 31

12.3 Uses

A discrete structure is often used as the “default structure” on a set that doesn't carry any other natural topology,uniformity, or metric; discrete structures can often be used as “extreme” examples to test particular suppositions.For example, any group can be considered as a topological group by giving it the discrete topology, implying thattheorems about topological groups apply to all groups. Indeed, analysts may refer to the ordinary, non-topologicalgroups studied by algebraists as "discrete groups" . In some cases, this can be usefully applied, for example incombination with Pontryagin duality. A 0-dimensional manifold (or differentiable or analytical manifold) is nothingbut a discrete topological space. We can therefore view any discrete group as a 0-dimensional Lie group.A product of countably infinite copies of the discrete space of natural numbers is homeomorphic to the space ofirrational numbers, with the homeomorphism given by the continued fraction expansion. A product of countablyinfinite copies of the discrete space 0,1 is homeomorphic to the Cantor set; and in fact uniformly homeomorphicto the Cantor set if we use the product uniformity on the product. Such a homeomorphism is given by using ternarynotation of numbers. (See Cantor space.)In the foundations of mathematics, the study of compactness properties of products of 0,1 is central to the topo-logical approach to the ultrafilter principle, which is a weak form of choice.

12.4 Indiscrete spaces

Main article: Trivial topology

In some ways, the opposite of the discrete topology is the trivial topology (also called the indiscrete topology), whichhas the fewest possible open sets (just the empty set and the space itself). Where the discrete topology is initialor free, the indiscrete topology is final or cofree: every function from a topological space to an indiscrete space iscontinuous, etc.

12.5 See also• Cylinder set

• Taxicab geometry

12.6 References[1] Pleasants, Peter A.B. (2000). “Designer quasicrystals: Cut-and-project sets with pre-assigned properties”. In Baake,

Michael. Directions in mathematical quasicrystals. CRM Monograph Series 13. Providence, RI: American MathematicalSociety. pp. 95–141. ISBN 0-8218-2629-8. Zbl 0982.52018.

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1978). Counterexamples in Topology (2nd ed.). Berlin, New York:Springer-Verlag. ISBN 3-540-90312-7. MR 507446. Zbl 0386.54001.

Chapter 13

Discrete two-point space

In topology, a branch of mathematics, a discrete two-point space is the simplest example of a totally disconnecteddiscrete space. The points can be denoted by the symbols 0 and 1.Any disconnected space has a continuous mapping onto the discrete two-point space. Conversely if a continuousmapping to the discrete two-point space exists from a topological space, the space is disconnected.[1]

13.1 References[1] George F. Simmons (1968). Introduction to Topology and Modern Analysis. McGraw–Hill Book Company. p. 144.

32

Chapter 14

Dogbone space

The first stage of the dogbone space construction.

In geometric topology, the dogbone space, constructed by R. H. Bing (1957), is a quotient space of three-dimensionalEuclidean space R3 such that all inverse images of points are points or tame arcs, yet it is not homeomorphic to R3.The name “dogbone space” refers to a fanciful resemblance between some of the diagrams of genus 2 surfaces in R.H.Bing’s paper and a dog bone. Bing (1959) showed that the product of the dogbone space with R1 is homeomorphicto R4.Although the dogbone space is not a manifold, it is a generalized homological manifold and a homotopy manifold.

14.1 See also

• Whitehead manifold, a 3-manifold not homeomorphic to R3 whose product with R1 is homeomorphic to R4.

14.2 References

• Daverman, Robert J. (2007), Decompositions of manifolds, AMS Chelsea Publishing, Providence, RI, p. 22,ISBN 978-0-8218-4372-7, MR 2341468

• Bing, R. H. (1957), “A decomposition of E3 into points and tame arcs such that the decomposition spaceis topologically different from E3", Annals of Mathematics. Second Series 65: 484–500, ISSN 0003-486X,JSTOR 1970058, MR 0092961

33

34 CHAPTER 14. DOGBONE SPACE

• Bing, R. H. (1959), “The cartesian product of a certain nonmanifold and a line is E4", Annals of Mathematics.Second Series 70: 399–412, ISSN 0003-486X, JSTOR 1970322, MR 0107228

Chapter 15

Dunce hat (topology)

To get a dunce hat, take a solid triangle and successively glue together all three sides with the indicated orientation.

In topology, the dunce hat is a compact topological space formed by taking a solid triangle and gluing all three sidestogether, with the orientation of one side reversed. Simply gluing two sides oriented in the same direction would yielda cone much like the layman’s dunce cap, but the gluing of the third side results in identifying the base of the capwith a line joining the base to the point.The dunce hat is contractible, but not collapsible. Contractibility can be easily seen by noting that the dunce hatembeds in the 3-ball and the 3-ball deformation retracts onto the dunce hat. Alternatively, note that the dunce hat

35

36 CHAPTER 15. DUNCE HAT (TOPOLOGY)

is the CW-complex obtained by gluing the boundary of a 2-cell onto the circle. The gluing map is homotopic to theidentity map on the circle and so the complex is homotopy equivalent to the disc. By contrast, it is not collapsiblebecause it does not have a free face.The name is due to E. C. Zeeman, who observed that any contractible 2-complex (such as the dunce hat) after takingthe Cartesian product with the closed unit interval seemed to be collapsible. This observation became known as theZeeman conjecture and was shown by Zeeman to imply the Poincaré conjecture.

Dunce hat Folding. The blue hole is only for better view: it may be filled by a spherical cap. The (green) triangle border folds on acircle.

15.1 See also• House with two rooms

15.2 References• Zeeman, E. C., On the dunce hat, Topology 2 (1964), 341–358.

Chapter 16

Equivariant topology

In mathematics, equivariant topology is the study of topological spaces with group actions.

16.1 See also• Equivariant cohomology

• Mackey functor

• Equivariant stable homotopy theory

• G-spectrum

37

Chapter 17

Erdős space

In mathematics, Erdős space is a topological space named after Paul Erdős.Erdős space is defined as the set E of points in the Hilbert space l2 of square summable sequences having allcoordinates rational. Erdős space is a totally disconnected, one-dimensional topological space. The space E ishomeomorphic to the direct product E×E. Endowed with the compact-open topology, the set of all homeomorphismsof the Euclidean space Rn leaving the set Qn of vectors with rational coordinates invariant is homeomorphic to theErdős space for n ≥ 2.[1]

17.1 References[1] Jan J. Dijkstra, Jan van Mill. Erdős Space and Homeomorphism Groups of Manifolds. Memoirs of the American Mathe-

matical Society, Number 979.

38

Chapter 18

Euclidean space

This article is about Euclidean spaces of all dimensions. For 3-dimensional Euclidean space, see 3-dimensionalspace.In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space

of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid ofAlexandria.[1] The term “Euclidean” distinguishes these spaces from other types of spaces considered in moderngeometry. Euclidean spaces also generalize to higher dimensions.Classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates,while the other properties of these spaces were deduced as theorems. Geometric constructions are also used to definerational numbers. When algebra and mathematical analysis became developed enough, this relation reversed and nowit is more common to define Euclidean space using Cartesian coordinates and the ideas of analytic geometry. It meansthat points of the space are specified with collections of real numbers, and geometric shapes are defined as equationsand inequalities. This approach brings the tools of algebra and calculus to bear on questions of geometry and has theadvantage that it generalizes easily to Euclidean spaces of more than three dimensions.From the modern viewpoint, there is essentially only one Euclidean space of each dimension. With Cartesian coor-dinates it is modelled by the real coordinate space (Rn) of the same dimension. In one dimension, this is the realline; in two dimensions, it is the Cartesian plane; and in higher dimensions it is a coordinate space with three or morereal number coordinates. Mathematicians denote the n-dimensional Euclidean space by En if they wish to emphasizeits Euclidean nature, but Rn is used as well since the latter is assumed to have the standard Euclidean structure, andthese two structures are not always distinguished. Euclidean spaces have finite dimension.[2]

18.1 Intuitive overview

One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms ofdistance and angle. For example, there are two fundamental operations (referred to as motions) on the plane. One istranslation, which means a shifting of the plane so that every point is shifted in the same direction and by the samedistance. The other is rotation about a fixed point in the plane, in which every point in the plane turns about that fixedpoint through the same angle. One of the basic tenets of Euclidean geometry is that two figures (usually consideredas subsets) of the plane should be considered equivalent (congruent) if one can be transformed into the other by somesequence of translations, rotations and reflections (see below).In order to make all of this mathematically precise, the theory must clearly define the notions of distance, angle,translation, and rotation for a mathematically described space. Even when used in physical theories, Euclidean spaceis an abstraction detached from actual physical locations, specific reference frames, measurement instruments, and soon. A purely mathematical definition of Euclidean space also ignores questions of units of length and other physicaldimensions: the distance in a “mathematical” space is a number, not something expressed in inches or metres. Thestandard way to define such space, as carried out in the remainder of this article, is to define the Euclidean plane asa two-dimensional real vector space equipped with an inner product.[2] The reason for working with arbitrary vectorspaces instead of Rn is that it is often preferable to work in a coordinate-free manner (that is, without choosing apreferred basis). For then:

• the vectors in the vector space correspond to the points of the Euclidean plane,

39

40 CHAPTER 18. EUCLIDEAN SPACE

A sphere, the most perfect spatial shape according to Pythagoreans, also is an important concept in modern understanding of Eu-clidean spaces

• the addition operation in the vector space corresponds to translation, and

• the inner product implies notions of angle and distance, which can be used to define rotation.

Once the Euclidean plane has been described in this language, it is actually a simple matter to extend its concept toarbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficultby the presence of more dimensions. (However, rotations are more subtle in high dimensions, and visualizing high-dimensional spaces remains difficult, even for experienced mathematicians.)A Euclidean space is not technically a vector space but rather an affine space, on which a vector space acts by transla-tions, or, conversely, a Euclidean vector is the difference (displacement) in an ordered pair of points, not a single point.Intuitively, the distinction says merely that there is no canonical choice of where the origin should go in the space,because it can be translated anywhere. When a certain point is chosen, it can be declared the origin and subsequentcalculations may ignore the difference between a point and its coordinate vector, as said above. See point–vectordistinction for details.

18.2. EUCLIDEAN STRUCTURE 41

X

Y

Z

O

xy

z

(x,y,z)

Every point in three-dimensional Euclidean space is determined by three coordinates.

18.2 Euclidean structure

These are distances between points and the angles between lines or vectors, which satisfy certain conditions (seebelow), which makes a set of points a Euclidean space. The natural way to obtain these quantities is by introducingand using the standard inner product (also known as the dot product) on Rn.[2] The inner product of any two realn-vectors x and y is defined by

x · y =n∑

i=1

xiyi = x1y1 + x2y2 + · · ·+ xnyn,

where xᵢ and yᵢ are ith coordinates of vectors x and y respectively. The result is always a real number.

18.2.1 Distance

Main article: Euclidean distance

The inner product of x with itself is always non-negative. This product allows us to define the “length” of a vector xthrough square root:

42 CHAPTER 18. EUCLIDEAN SPACE

∥x∥ =√x · x =

√√√√ n∑i=1

(xi)2.

This length function satisfies the required properties of a norm and is called the Euclidean norm on Rn.Finally, one can use the norm to define a metric (or distance function) on Rn by

d(x, y) = ∥x− y∥ =

√√√√ n∑i=1

(xi − yi)2.

This distance function is called the Euclidean metric. This formula expresses a special case of the Pythagoreantheorem.This distance function (which makes a metric space) is sufficient to define all Euclidean geometry, including the dotproduct. Thus, a real coordinate space together with this Euclidean structure is called Euclidean space. Its vectorsform an inner product space (in fact a Hilbert space), and a normed vector space.The metric space structure is the main reason behind the use of real numbers R, not some other ordered field, asthe mathematical foundation of Euclidean (and many other) spaces. Euclidean space is a complete metric space, aproperty which is impossible to achieve operating over rational numbers, for example.

18.2.2 Angle

Main article: AngleThe (non-reflex) angle θ (0° ≤ θ ≤ 180°) between vectors x and y is then given by

θ = arccos( x · y∥x∥∥y∥

)where arccos is the arccosine function. It is useful only for n > 1,[footnote 1] and the case n = 2 is somewhat special.Namely, on an oriented Euclidean plane one can define an angle between two vectors as a number defined modulo 1turn (usually denoted as either 2π or 360°), such that ∠y x = −∠x y. This oriented angle is equal either to the angleθ from the formula above or to −θ. If one non-zero vector is fixed (such as the first basis vector), then each non-zerovector is uniquely defined by its magnitude and angle.The angle does not change if vectors x and y are multiplied by positive numbers.Unlike the aforementioned situation with distance, the scale of angles is the same in pure mathematics, physics, andcomputing. It does not depend on the scale of distances; all distances may be multiplied by some fixed factor, andall angles will be preserved. Usually, the angle is considered a dimensionless quantity, but there are different unitsof measurement, such as radian (preferred in pure mathematics and theoretical physics) and degree (°) (preferred inmost applications).

18.2.3 Rotations and reflections

Main articles: Rotation (mathematics), Reflection (mathematics) and Orthogonal groupSee also: rotational symmetry and reflection symmetry

Symmetries of a Euclidean space are transformations which preserve the Euclidean metric (called isometries). Al-though aforementioned translations are most obvious of them, they have the same structure for any affine space anddo not show a distinctive character of Euclidean geometry. Another family of symmetries leave one point fixed,which may be seen as the origin without loss of generality. All transformations, which preserves the origin and theEuclidean metric, are linear maps. Such transformations Q must, for any x and y, satisfy:

18.2. EUCLIDEAN STRUCTURE 43

45°-315°405°

Positive and negative angles on the oriented plane

Qx ·Qy = x · y (explain the notation),|Qx| = |x|.

Such transforms constitute a group called the orthogonal group O(n). Its elements Q are exactly solutions of a matrixequation

QTQ = QQT = I,

where QT is the transpose of Q and I is the identity matrix.But a Euclidean space is orientable.[footnote 2] Each of these transformations either preserves or reverses orientationdepending on whether its determinant is +1 or −1 respectively. Only transformations which preserve orientation,which form the special orthogonal group SO(n), are considered (proper) rotations. This group has, as a Lie group,the same dimension n(n − 1) /2 and is the identity component of O(n).

44 CHAPTER 18. EUCLIDEAN SPACE

Groups SO(n) are well-studied for n ≤ 4. There are no non-trivial rotations in 0- and 1-spaces. Rotations of aEuclidean plane (n = 2) are parametrized by the angle (modulo 1 turn). Rotations of a 3-space are parametrized withaxis and angle, whereas a rotation of a 4-space is a superposition of two 2-dimensional rotations around perpendicularplanes.Among linear transforms in O(n) which reverse the orientation are hyperplane reflections. This is the only possiblecase for n ≤ 2, but starting from three dimensions, such isometry in the general position is a rotoreflection.

18.2.4 Euclidean group

Main article: Euclidean group

The Euclidean group E(n), also referred to as the group of all isometries ISO(n), treats translations, rotations, andreflections in a uniform way, considering them as group actions in the context of group theory, and especially in Liegroup theory. These group actions preserve the Euclidean structure.As the group of all isometries, ISO(n), the Euclidean group is important because it makes Euclidean geometry a caseof Klein geometry, a theoretical framework including many alternative geometries.The structure of Euclidean spaces – distances, lines, vectors, angles (up to sign), and so on – is invariant under thetransformations of their associated Euclidean group. For instance, translations form a commutative subgroup thatacts freely and transitively on En, while the stabilizer of any point there is the aforementioned O(n).Along with translations, rotations, reflections, as well as the identity transformation, Euclidean motions comprise alsoglide reflections (for n ≥ 2), screw operations and rotoreflections (for n ≥ 3), and even more complex combinationsof primitive transformations for n ≥ 4.The group structure determines which conditions a metric space needs to satisfy to be a Euclidean space:

1. Firstly, a metric space must be translationally invariant with respect to some (finite-dimensional) real vectorspace. This means that the space itself is an affine space, that the space is flat, not curved, and points do nothave different properties, and so any point can be translated to any other point.

2. Secondly, the metric must correspond in the aforementioned way to some positive-defined quadratic form onthis vector space, because point stabilizers have to be isomorphic to O(n).

18.3 Non-Cartesian coordinates

Main article: Coordinate system

Cartesian coordinates are arguably the standard, but not the only possible option for a Euclidean space. Skew coordi-nates are compatible with the affine structure of En, but make formulae for angles and distances more complicated.Another approach, which goes in line with ideas of differential geometry and conformal geometry, is orthogonalcoordinates, where coordinate hypersurfaces of different coordinates are orthogonal, although curved. Examplesinclude the polar coordinate system on Euclidean plane, the second important plane coordinate system.See below about expression of the Euclidean structure in curvilinear coordinates.

18.4 Geometric shapes

See also: List of mathematical shapes

18.4. GEOMETRIC SHAPES 45

3-dimensional skew coordinates

Parabolic coordinates

18.4.1 Lines, planes, and other subspaces

Main article: Flat (geometry)

The simplest (after points) objects in Euclidean space are flats, or Euclidean subspaces of lesser dimension. Pointsare 0-dimensional flats, 1-dimensional flats are called (straight) lines, and 2-dimensional flats are planes. (n − 1)-dimensional flats are called hyperplanes.

46 CHAPTER 18. EUCLIDEAN SPACE

Barycentric coordinates in 3-dimensional space: four coordinates are related with one linear equation

Any two distinct points lie on exactly one line. Any line and a point outside it lie on exactly one plane. Moregenerally, the properties of flats and their incidence of Euclidean space are shared with affine geometry, whereas theaffine geometry is devoid of distances and angles.

18.4.2 Line segments and triangles

Main articles: Line segment and Triangle geometry

This is not only a line which a pair (A, B) of distinct points defines. Points on the line which lie between A and B,together with A and B themselves, constitute a line segment A B. Any line segment has the length, which equals todistance between A and B. If A = B, then the segment is degenerate and its length equals to 0, otherwise the lengthis positive.A (non-degenerate) triangle is defined by three points not lying on the same line. Any triangle lies on one plane. Theconcept of triangle is not specific to Euclidean spaces, but Euclidean triangles have numerous special properties anddefine many derived objects.A triangle can be thought of as a 3-gon on a plane, a special (and the first meaningful in Euclidean geometry) case ofa polygon.

18.4. GEOMETRIC SHAPES 47

Three mutually transversal planes in the 3-dimensional space and their intersections, three lines

18.4.3 Polytopes and root systems

Main articles: Polytope and Root systemSee also: List of polygons, polyhedra and polytopes and List of regular polytopes

Polytope is a concept that generalizes polygons on a plane and polyhedra in 3-dimensional space (which are amongthe earliest studied geometrical objects). A simplex is a generalization of a line segment (1-simplex) and a triangle(2-simplex). A tetrahedron is a 3-simplex.The concept of a polytope belongs to affine geometry, which is more general than Euclidean. But Euclidean geometrydistinguish regular polytopes. For example, affine geometry does not see the difference between an equilateral triangleand a right triangle, but in Euclidean space the former is regular and the latter is not.Root systems are special sets of Euclidean vectors. A root system is often identical to the set of vertices of a regularpolytope.

18.4.4 Curves

Main article: Euclidean geometry of curvesSee also: List of curves

18.4.5 Balls, spheres, and hypersurfaces

Main articles: Ball (mathematics) and HypersurfaceSee also: n-sphere and List of surfaces

48 CHAPTER 18. EUCLIDEAN SPACE

18.5 Topology

Main article: Real coordinate space § Topological properties

Since Euclidean space is a metric space, it is also a topological space with the natural topology induced by the metric.The metric topology on En is called the Euclidean topology, and it is identical to the standard topology on Rn. Aset is open if and only if it contains an open ball around each of its points; in other words, open balls form a baseof the topology. The topological dimension of the Euclidean n-space equals n, which implies that spaces of differentdimension are not homeomorphic. A finer result is the invariance of domain, which proves that any subset of n-space,that is (with its subspace topology) homeomorphic to an open subset of n-space, is itself open.

18.6 Applications

Aside from countless uses in fundamental mathematics, a Euclidean model of the physical space can be used to solvemany practical problems with sufficient precision. Two usual approaches are a fixed, or stationary reference frame(i.e. the description of a motion of objects as their positions that change continuously with time), and the use ofGalilean space-time symmetry (such as in Newtonian mechanics). To both of them the modern Euclidean geometryprovides a convenient formalism; for example, the space of Galilean velocities is itself a Euclidean space (see relativevelocity for details).Topographical maps and technical drawings are planar Euclidean. An idea behind them is the scale invariance ofEuclidean geometry, that permits to represent large objects in a small sheet of paper, or a screen.

18.7 Alternatives and generalizations

Although Euclidean spaces are no longer considered to be the only possible setting for a geometry, they act as pro-totypes for other geometric objects. Ideas and terminology from Euclidean geometry (both traditional and analytic)are pervasive in modern mathematics, where other geometric objects share many similarities with Euclidean spaces,share part of their structure, or embed Euclidean spaces.

18.7.1 Curved spaces

Main article: Riemannian geometry

A smooth manifold is a Hausdorff topological space that is locally diffeomorphic to Euclidean space. Diffeomor-phism does not respect distance and angle, but if one additionally prescribes a smoothly varying inner product onthe manifold’s tangent spaces, then the result is what is called a Riemannian manifold. Put differently, a Riemannianmanifold is a space constructed by deforming and patching together Euclidean spaces. Such a space enjoys notionsof distance and angle, but they behave in a curved, non-Euclidean manner. The simplest Riemannian manifold, con-sisting of Rn with a constant inner product, is essentially identical to Euclidean n-space itself. Less trivial examplesare n-sphere and hyperbolic spaces. Discovery of the latter in the 19th century was branded as the non-Euclideangeometry.Also, the concept of a Riemannianmanifold permits an expression of the Euclidean structure in any smooth coordinatesystem, via metric tensor. From this tensor one can compute the Riemann curvature tensor. Where the latter equals tozero, the metric structure is locally Euclidean (it means that at least some open set in the coordinate space is isometricto a piece of Euclidean space), no matter whether coordinates are affine or curvilinear.

18.7.2 Indefinite quadratic form

See also: Sylvester’s law of inertia

18.8. SEE ALSO 49

If one replaces the inner product of a Euclidean space with an indefinite quadratic form, the result is a pseudo-Euclidean space. Smooth manifolds built from such spaces are called pseudo-Riemannian manifolds. Perhaps theirmost famous application is the theory of relativity, where flat spacetime is a pseudo-Euclidean space calledMinkowskispace, where rotations correspond to motions of hyperbolic spaces mentioned above. Further generalization to curvedspacetimes form pseudo-Riemannian manifolds, such as in general relativity.

18.7.3 Other number fields

Another line of generalization is to consider other number fields than one of real numbers. Over complex numbers, aHilbert space can be seen as a generalization of Euclidean dot product structure, although the definition of the innerproduct becomes a sesquilinear form for compatibility with metric structure.

18.7.4 Infinite dimensions

Main articles: inner product space and Hilbert space

18.8 See also• Function of several real variables, a coordinate presentation of a function on a Euclidean space

• Geometric algebra, an alternative algebraic formalism

• Vector calculus, a standard algebraic formalism

18.9 Footnotes[1] On the real line (n = 1) any two non-zero vectors are either parallel or antiparallel depending on whether their signs match

or oppose. There are no angles between 0 and 180°.

[2] It is Rn which is oriented because of the ordering of elements of the standard basis. Although an orientation is not anattribute of the Euclidean structure, there are only two possible orientations, and any linear automorphism either keepsorientation or reverses (swaps the two).

18.10 References[1] Ball, W.W. Rouse (1960) [1908]. A Short Account of the History of Mathematics (4th ed.). Dover Publications. pp. 50–62.

ISBN 0-486-20630-0.

[2] E.D. Solomentsev (7 February 2011). “Euclidean space.”. Encyclopedia of Mathematics. Springer. Retrieved 1 May 2014.

18.11 External links• Hazewinkel, Michiel, ed. (2001), “Euclidean space”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 19

Excluded point topology

In mathematics, the excluded point topology is a topology where exclusion of a particular point defines openness.Formally, let X be any set and p ∈ X. The collection

T = S ⊆ X: p ∉ S or S = X;

of subsets of X is then the excluded point topology on X. There are a variety of cases which are individually named:

• If X has two points we call it the Sierpiński space. This case is somewhat special and is handled separately.

• If X is finite (with at least 3 points) we call the topology on X the finite excluded point topology

• If X is countably infinite we call the topology on X the countable excluded point topology

• If X is uncountable we call the topology on X the uncountable excluded point topology

A generalization / related topology is the open extension topology. That is if X\p has the discrete topology thenthe open extension topology will be the excluded point topology.This topology is used to provide interesting examples and counterexamples. Excluded point topology is also connectedand that is clear since the only open set containing the excluded point is X itself and hence X cannot be written asdisjoint union of two proper open subsets.

19.1 See also• Sierpiński space

• Particular point topology

• Alexandrov topology

• Finite topological space

• Fort space

19.2 References• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446

my notes Taha el Turki.[1]

50

Chapter 20

Extension topology

In topology, a branch ofmathematics, an extension topology is a topology placed on the disjoint union of a topologicalspace and another set.There are various types of extension topology, described in the sections below.

20.1 Extension topology

Let X be a topological space and P a set disjoint from X. Consider in X ∪ P the topology whose open sets are of theform: A ∪ Q, where A is an open set of X and Q is a subset of P.Note that the closed sets of X ∪ P are of the form: B ∪ Q, where B is a closed set of X and Q is a subset of P.For these reasons this topology is called the extension topology of X plus P, with which one extends to X ∪ P theopen and the closed sets of X. Note that the subspace topology of X as a subset of X ∪ P is the original topology ofX, while the subspace topology of P as a subset of X ∪ P is the discrete topology.Being Y a topological space and R a subset of Y, one might ask whether the extension topology of Y - R plus R isthe same as the original topology of Y, and the answer is in general no.Note the similitude of this extension topology construction and the Alexandroff one-point compactification, in whichcase, having a topological space X which one wishes to compactify by adding a point ∞ in infinity, one considers theclosed sets of X ∪ ∞ to be the sets of the form: K, where K is a closed compact set of X, or B ∪ ∞, where B isa closed set of X.

20.2 Open extension topology

Let X be a topological space and P a set disjoint from X. Consider in X ∪ P the topology whose open sets are of theform: X ∪ Q, where Q is a subset of P, or A, where A is an open set of X.For this reason this topology is called the open extension topology of X plus P, with which one extends to X ∪ Pthe open sets of X. Note that the subspace topology of X as a subset of X ∪ P is the original topology of X, while thesubspace topology of P as a subset of X ∪ P is the discrete topology.Note that the closed sets of X ∪ P are of the form: Q, where Q is a subset of P, or B ∪ P, where B is a closed set ofX.Being Y a topological space and R a subset of Y, one might ask whether the extension topology of Y - R plus R isthe same as the original topology of Y, and the answer is in general no.Note that the open extension topology of X ∪ P is smaller than the extension topology of X ∪ P.Being Z a set and p a point in Z, one obtains the excluded point topology construction by considering in Z the discretetopology and applying the open extension topology construction to Z - p plus p.

51

52 CHAPTER 20. EXTENSION TOPOLOGY

20.3 Closed extension topology

Let X be a topological space and P a set disjoint from X. Consider in X ∪ P the topology whose closed sets are of theform: X ∪ Q, where Q is a subset of P, or B, where B is a closed set of X.For this reason this topology is called the closed extension topology of X plus P, with which one extends to X ∪ Pthe closed sets of X. Note that the subspace topology of X as a subset of X ∪ P is the original topology of X, whilethe subspace topology of P as a subset of X ∪ P is the discrete topology.Note that the open sets of X ∪ P are of the form: Q, where Q is a subset of P, or A ∪ P, where A is an open set of X.Being Y a topological space and R a subset of Y, one might ask whether the extension topology of Y - R plus R isthe same as the original topology of Y, and the answer is in general no.Note that the closed extension topology of X ∪ P is smaller than the extension topology of X ∪ P.Being Z a set and p a point in Z, one obtains the particular point topology construction by considering in Z the discretetopology and applying the closed extension topology construction to Z - p plus p.

20.4 References• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446

Chapter 21

Finite topological space

In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is,it is a topological space for which there are only finitely many points.While topology has mainly been developed for infinite spaces, finite topological spaces are often used to provide ex-amples of interesting phenomena or counterexamples to plausible sounding conjectures. William Thurston has calledthe study of finite topologies in this sense “an oddball topic that can lend good insight to a variety of questions.”[1]

21.1 Topologies on a finite set

21.1.1 As a bounded sublattice

A topology on a set X is defined as a subset of P(X), the power set of X, which includes both ∅ and X and is closedunder finite intersections and arbitrary unions.Since the power set of a finite set is finite there can be only finitely many open sets (and only finitely many closedsets). Therefore one only need check that the union of a finite number of open sets is open. This leads to a simplerdescription of topologies on a finite set.Let X be a finite set. A topology on X is a subset τ of P(X) such that

1. ∅ ∈ τ and X ∈ τ

2. if U and V are in τ then U ∪ V ∈ τ

3. if U and V are in τ then U ∩ V ∈ τ

A topology on a finite set is therefore nothing more than a sublattice of (P(X), ⊂) which includes both the bottomelement (∅) and the top element (X).Every finite bounded lattice is complete since the meet or join of any family of elements can always be reduced toa meet or join of two elements. It follows that in a finite topological space the union or intersection of an arbitraryfamily of open sets (resp. closed sets) is open (resp. closed).

21.1.2 Specialization preorder

Topologies on a finite set X are in one-to-one correspondence with preorders on X. Recall that a preorder on X is abinary relation on X which is reflexive and transitive.Given a (not necessarily finite) topological space X we can define a preorder on X by

x ≤ y if and only if x ∈ cly

53

54 CHAPTER 21. FINITE TOPOLOGICAL SPACE

where cly denotes the closure of the singleton set y. This preorder is called the specialization preorder on X.Every open set U of X will be an upper set with respect to ≤ (i.e. if x ∈ U and x ≤ y then y ∈ U). Now if X is finite,the converse is also true: every upper set is open in X. So for finite spaces, the topology on X is uniquely determinedby ≤.Going in the other direction, suppose (X, ≤) is a preordered set. Define a topology τ on X by taking the open sets tobe the upper sets with respect to ≤. Then the relation ≤ will be the specialization preorder of (X, τ). The topologydefined in this way is called the Alexandrov topology determined by ≤.The equivalence between preorders and finite topologies can be interpreted as a version of Birkhoff’s representationtheorem, an equivalence between finite distributive lattices (the lattice of open sets of the topology) and partial orders(the partial order of equivalence classes of the preorder). This correspondence also works for a larger class of spacescalled finitely generated spaces. Finitely generated spaces can be characterized as the spaces in which an arbitraryintersection of open sets is open. Finite topological spaces are a special class of finitely generated spaces.

21.2 Examples

21.2.1 0 or 1 points

There is a unique topology on the empty set ∅. The only open set is the empty one. Indeed, this is the only subset of∅.Likewise, there is a unique topology on a singleton set a. Here the open sets are ∅ and a. This topology is bothdiscrete and trivial, although in some ways it is better to think of it as a discrete space since it shares more propertieswith the family of finite discrete spaces.For any topological space X there is a unique continuous function from ∅ to X, namely the empty function. Thereis also a unique continuous function from X to the singleton space a, namely the constant function to a. In thelanguage of category theory the empty space serves as an initial object in the category of topological spaces while thesingleton space serves as a terminal object.

21.2.2 2 points

Let X = a,b be a set with 2 elements. There are four distinct topologies on X:

1. ∅, a,b (the trivial topology)

2. ∅, a, a,b

3. ∅, b, a,b

4. ∅, a, b, a,b (the discrete topology)

The second and third topologies above are easily seen to be homeomorphic. The function from X to itself whichswaps a and b is a homeomorphism. A topological space homeomorphic to one of these is called a Sierpiński space.So, in fact, there are only three inequivalent topologies on a two-point set: the trivial one, the discrete one, and theSierpiński topology.The specialization preorder on the Sierpiński space a,b with b open is given by: a ≤ a, b ≤ b, and a ≤ b.

21.2.3 3 points

Let X = a,b,c be a set with 3 elements. There are 29 distinct topologies on X but only 9 inequivalent topologies:

1. ∅, a,b,c

2. ∅, c, a,b,c

3. ∅, a,b, a,b,c

21.3. PROPERTIES 55

4. ∅, c, a,b, a,b,c

5. ∅, c, b,c, a,b,c

6. ∅, c, a,c, b,c, a,b,c

7. ∅, a, b, a,b, a,b,c

8. ∅, b, c, a,b, b,c, a,b,c

9. ∅, a, b, c, a,b, a,c, b,c, a,b,c

The last 5 of these are all T0. The first one is trivial, while in 2, 3, and 4 the points a and b are topologicallyindistinguishable.

21.3 Properties

21.3.1 Compactness and countability

Every finite topological space is compact since any open cover must already be finite. Indeed, compact spaces areoften thought of as a generalization of finite spaces since they share many of the same properties.Every finite topological space is also second-countable (there are only finitely many open sets) and separable (sincethe space itself is countable).

21.3.2 Separation axioms

If a finite topological space is T1 (in particular, if it is Hausdorff) then it must, in fact, be discrete. This is becausethe complement of a point is a finite union of closed points and therefore closed. It follows that each point must beopen.Therefore, any finite topological space which is not discrete cannot be T1, Hausdorff, or anything stronger.However, it is possible for a non-discrete finite space to be T0. In general, two points x and y are topologicallyindistinguishable if and only if x ≤ y and y ≤ x, where ≤ is the specialization preorder on X. It follows that a space Xis T0 if and only if the specialization preorder ≤ on X is a partial order. There are numerous partial orders on a finiteset. Each defines a unique T0 topology.Similarly, a space is R0 if and only if the specialization preorder is an equivalence relation. Given any equivalencerelation on a finite set X the associated topology is the partition topology on X. The equivalence classes will be theclasses of topologically indistinguishable points. Since the partition topology is pseudometrizable, a finite space is R0

if and only if it is completely regular.Non-discrete finite spaces can also be normal. The excluded point topology on any finite set is a completely normalT0 space which is non-discrete.

21.3.3 Connectivity

Connectivity in a finite spaceX is best understood by considering the specialization preorder ≤ onX. We can associateto any preordered set X a directed graph Γ by taking the points of X as vertices and drawing an edge x→ y wheneverx ≤ y. The connectivity of a finite space X can be understood by considering the connectivity of the associated graphΓ.In any topological space, if x ≤ y then there is a path from x to y. One can simply take f(0) = x and f(t) = y for t > 0.It is easily to verify that f is continuous. It follows that the path components of a finite topological space are preciselythe (weakly) connected components of the associated graph Γ. That is, there is a topological path from x to y if andonly if there is an undirected path between the corresponding vertices of Γ.Every finite space is locally path-connected since the set

56 CHAPTER 21. FINITE TOPOLOGICAL SPACE

↑x = y ∈ X : x ≤ y

is a path-connected open neighborhood of x that is contained in every other neighborhood. In other words, this singleset forms a local base at x.Therefore, a finite space is connected if and only if it is path-connected. The connected components are precisely thepath components. Each such component is both closed and open in X.Finite spaces may have stronger connectivity properties. A finite space X is

• hyperconnected if and only if there is a greatest element with respect to the specialization preorder. This is anelement whose closure is the whole space X.

• ultraconnected if and only if there is a least element with respect to the specialization preorder. This is anelement whose only neighborhood is the whole space X.

For example, the particular point topology on a finite space is hyperconnected while the excluded point topology isultraconnected. The Sierpiński space is both.

21.3.4 Additional structure

A finite topological space is pseudometrizable if and only if it is R0. In this case, one possible pseudometric is givenby

d(x, y) =

0 x ≡ y

1 x ≡ y

where x ≡ y means x and y are topologically indistinguishable. A finite topological space is metrizable if and only ifit is discrete.Likewise, a topological space is uniformizable if and only if it is R0. The uniform structure will be the pseudometricuniformity induced by the above pseudometric.

21.3.5 Algebraic topology

Perhaps surprisingly, there are finite topological spaces with nontrivial fundamental groups. A simple example is thepseudocircle, which is space X with four points, two of which are open and two of which are closed. There is acontinuous map from the unit circle S1 to X which is a weak homotopy equivalence (i.e. it induces an isomorphismof homotopy groups). It follows that the fundamental group of the pseudocircle is infinite cyclic.More generally it has been shown that for any finite abstract simplicial complex K, there is a finite topological spaceXK and a weak homotopy equivalence f : |K | → XK where |K | is the geometric realization of K. It follows that thehomotopy groups of |K | and XK are isomorphic. In fact, the underlying set of XK can be take to be K itself, with thetopology associated to the inclusion partial order.

21.4 Number of topologies on a finite set

As discussed above, topologies on a finite set are in one-to-one correspondence with preorders on the set, and T0

topologies are in one-to-one correspondence with partial orders. Therefore the number of topologies on a finite setis equal to the number of preorders and the number of T0 topologies is equal to the number of partial orders.The table below lists the number of distinct (T0) topologies on a set with n elements. It also lists the number ofinequivalent (i.e. nonhomeomorphic) topologies.Let T(n) denote the number of distinct topologies on a set with n points. There is no known simple formula tocompute T(n) for arbitrary n. The Online Encyclopedia of Integer Sequences presently lists T(n) for n ≤ 18.

21.5. SEE ALSO 57

The number of distinct T0 topologies on a set with n points, denoted T0(n), is related to T(n) by the formula

T (n) =n∑

k=0

S(n, k)T0(k)

where S(n,k) denotes the Stirling number of the second kind.

21.5 See also• Finite geometry

• Finite metric space

• Topological combinatorics

21.6 References[1] Thurston, William P. (April 1994). On Proof and Progress in Mathematics. Bulletin of the American Mathematical Society

30 (2). pp. 161–177. arXiv:math/9404236. doi:10.1090/S0273-0979-1994-00502-6.

• Finite topological spaces, RE Stong - Trans. Amer. Math. Soc, 1966

• Singular homology groups and homotopy groups of finite topological spaces, Michael C. McCord, Duke Math.J. Volume 33, Number 3 (1966), 465-474.

• Barmak, Jonathan (2011). Algebraic Topology of Finite Topological Spaces and Applications. Springer. ISBN978-3-642-22002-9.

• Merrifield, Richard; Simmons, Howard E. (1989). Topological Methods in Chemistry. Wiley. ISBN 978-0-471-83817-3.

21.7 External links• Notes and reading materials on finite topological spaces, J.P. MAY

Chapter 22

First uncountable ordinal

In mathematics, the first uncountable ordinal, traditionally denoted byω1 or sometimes byΩ, is the smallest ordinalnumber that, considered as a set, is uncountable. It is the supremum of all countable ordinals. The elements of ω1

are the countable ordinals, of which there are uncountably many.Like any ordinal number (in von Neumann’s approach), ω1 is a well-ordered set, with set membership ("∈") servingas the order relation. ω1 is a limit ordinal, i.e. there is no ordinal α with α + 1 = ω1.The cardinality of the set ω1 is the first uncountable cardinal number, ℵ1 (aleph-one). The ordinal ω1 is thus theinitial ordinal of ℵ1. Indeed, in most constructions ω1 and ℵ1 are equal as sets. To generalize: if α is an arbitraryordinal we define ωα as the initial ordinal of the cardinal ℵα.The existence of ω1 can be proven without the axiom of choice. (See Hartogs number.)

22.1 Topological properties

Any ordinal number can be turned into a topological space by using the order topology. When viewed as a topologicalspace, ω1 is often written as [0,ω1) to emphasize that it is the space consisting of all ordinals smaller than ω1.Every increasing ω-sequence of elements of [0,ω1) converges to a limit in [0,ω1). The reason is that the union(=supremum) of every countable set of countable ordinals is another countable ordinal.The topological space [0,ω1) is sequentially compact but not compact. As a consequence, it is not metrizable. It ishowever countably compact and thus not Lindelöf. In terms of axioms of countability, [0,ω1) is first countable butnot separable nor second countable.The space [0, ω1] = ω1 + 1 is compact and not first countable. ω1 is used to define the long line and the Tychonoffplank, two important counterexamples in topology.

22.2 See also• Ordinal arithmetic

• Large countable ordinal

• Continuum hypothesis

22.3 References• Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, ISBN3-540-44085-2.

• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York,1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).

58

Chapter 23

Fixed-point space

In mathematics, a Hausdorff space X is called a fixed-point space if every continuous function f : X → X has afixed point.For example, any closed interval [a,b] in R is a fixed point space, and it can be proved from the intermediate valueproperty of real continuous function. The open interval (a, b), however, is not a fixed point space. To see it, considerthe function f(x) = a+ 1

b−a · (x− a)2 , for example.Any linearly ordered space that is connected and has a top and a bottom element is a fixed point space.Note that, in the definition, we could easily have disposed of the condition that the space is Hausdorff.

23.1 References• Vasile I. Istratescu, Fixed Point Theory, An Introduction, D. Reidel, the Netherlands (1981). ISBN 90-277-1224-7

• Andrzej Granas and James Dugundji, Fixed Point Theory (2003) Springer-Verlag, New York, ISBN 0-387-00173-5

• William A. Kirk and Brailey Sims, Handbook of Metric Fixed Point Theory (2001), Kluwer Academic, LondonISBN 0-7923-7073-2

59

Chapter 24

Fort space

In mathematics, Fort space, named after M. K. Fort, Jr., is an example in the theory of topological spaces.Let X be an infinite set of points, of which P is one. Then a Fort space is defined by X together with all subsets Asuch that:

• A excludes P, or

• A contains all but a finite number of the points of X

X is homeomorphic to the one-point compactification of a discrete space.Modified Fort space is similar but has two particular points P and Q. So a subset is declared “open” if:

• A excludes P and Q, or

• A contains all but a finite number of the points of X

Fortissimo space is defined as follows. Let X be an uncountable set of points, of which P is one. A subset A isdeclared “open” if:

• A excludes P, or

• A contains all but a countable set of the points of X

24.1 See also• Arens–Fort space

• Appert topology

• Cofinite topology

• Excluded point topology

24.2 References• M. K. Fort, Jr. “Nested neighborhoods in Hausdorff spaces.” American Mathematical Monthly vol.62 (1955)372.

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446

60

Chapter 25

Geometric topology (object)

For the mathematical subject area, see geometric topology.

In mathematics, the geometric topology is a topology one can put on the set H of hyperbolic 3-manifolds of finitevolume.

25.1 Use

Convergence in this topology is a crucial ingredient of hyperbolic Dehn surgery, a fundamental tool in the theory ofhyperbolic 3-manifolds.

25.2 Definition

The following is a definition due to Troels Jorgensen:

A sequence Mi in H converges to M in H if there are

• a sequence of positive real numbers ϵi converging to 0, and• a sequence of (1 + ϵi) -bi-Lipschitz diffeomorphisms ϕi : Mi,[ϵi,∞) → M[ϵi,∞),

where the domains and ranges of the maps are the ϵi -thick parts of either theMi 's or M.

25.3 Alternate definition

There is an alternate definition due toMikhail Gromov. Gromov’s topology utilizes the Gromov-Hausdorffmetric andis defined on pointed hyperbolic 3-manifolds. One essentially considers better and better bi-Lipschitz homeomorphismson larger and larger balls. This results in the same notion of convergence as above as the thick part is always connected;thus, a large ball will eventually encompass all of the thick part.

25.3.1 On framed manifolds

As a further refinement, Gromov’s metric can also be defined on framed hyperbolic 3-manifolds. This gives nothingnew but this space can be explicitly identified with torsion-free Kleinian groups with the Chabauty topology.

25.4 See also• Algebraic topology (object)

61

62 CHAPTER 25. GEOMETRIC TOPOLOGY (OBJECT)

25.5 References• William Thurston, The geometry and topology of 3-manifolds, Princeton lecture notes (1978-1981).

• Canary, R. D.; Epstein, D. B. A.; Green, P., Notes on notes of Thurston. Analytical and geometric aspects ofhyperbolic space (Coventry/Durham, 1984), 3-−92, London Math. Soc. Lecture Note Ser., 111, CambridgeUniv. Press, Cambridge, 1987.

Chapter 26

Half-disk topology

In mathematics, and particularly general topology, the half-disk topology is an example of a topology given to theset X, given by all points (x,y) in the plane such that y ≥ 0.[1] The set X can be termed the closed upper half plane.To give the set X a topology means to say which subsets of X are “open”, and to do so in a way that the followingaxioms are met:[2]

1. The union of open sets is an open set.

2. The finite intersection of open sets is an open set.

3. The set X and the empty set ∅ are open sets.

26.1 Construction

We consider X to consist of the open upper half plane P, given by all points (x,y) in the plane such that y > 0; and thex-axis L, given by all points (x,y) in the plane such that y = 0. Clearly X is given by the union P ∪ L. The open upperhalf plane P has a topology given by the Euclidean metric topology.[1] We extend the topology on P to a topology onX = P ∪ L by adding some additional open sets. These extra sets are of the form (x,0) ∪ P ∩ U, where (x,0) isa point on the line L and U is an open, with respect to the Euclidean metric topology, neighbourhood of (x,y) in theplane.[1]

26.2 References[1] Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, pp. 96 – 97, ISBN 0-486-68735-X

[2] Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, p. 3, ISBN 0-486-68735-X

63

Chapter 27

Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topologicalspace in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on atopological space, the “Hausdorff condition” (T2) is the most frequently used and discussed. It implies the uniquenessof limits of sequences, nets, and filters.Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff’s original definitionof a topological space (in 1914) included the Hausdorff condition as an axiom.

27.1 Definitions

U

x

V

y

The points x and y, separated by their respective neighbourhoods U and V.

Points x and y in a topological space X can be separated by neighbourhoods if there exists a neighbourhood U of xand a neighbourhood V of y such that U and V are disjoint (U ∩ V = ∅). X is a Hausdorff space if any two distinctpoints of X can be separated by neighborhoods. This condition is the third separation axiom (after T0 and T1), whichis why Hausdorff spaces are also called T2 spaces. The name separated space is also used.A related, but weaker, notion is that of a preregular space. X is a preregular space if any two topologically distin-guishable points can be separated by neighbourhoods. Preregular spaces are also called R1 spaces.The relationship between these two conditions is as follows. A topological space is Hausdorff if and only if it is bothpreregular (i.e. topologically distinguishable points are separated by neighbourhoods) and Kolmogorov (i.e. distinctpoints are topologically distinguishable). A topological space is preregular if and only if its Kolmogorov quotient is

64

27.2. EQUIVALENCES 65

Hausdorff.

27.2 Equivalences

For a topological space X, the following are equivalent:

• X is a Hausdorff space.

• Limits of nets in X are unique.[1]

• Limits of filters on X are unique.[2]

• Any singleton set x ⊂ X is equal to the intersection of all closed neighbourhoods of x.[3] (A closed neigh-bourhood of x is a closed set that contains an open set containing x.)

• The diagonal Δ = (x,x) | x ∈ X is closed as a subset of the product space X × X.

27.3 Examples and counterexamples

Almost all spaces encountered in analysis are Hausdorff; most importantly, the real numbers (under the standardmetric topology on real numbers) are a Hausdorff space. More generally, all metric spaces are Hausdorff. In fact,many spaces of use in analysis, such as topological groups and topological manifolds, have the Hausdorff conditionexplicitly stated in their definitions.A simple example of a topology that is T1 but is not Hausdorff is the cofinite topology defined on an infinite set.Pseudometric spaces typically are not Hausdorff, but they are preregular, and their use in analysis is usually only in theconstruction of Hausdorff gauge spaces. Indeed, when analysts run across a non-Hausdorff space, it is still probablyat least preregular, and then they simply replace it with its Kolmogorov quotient, which is Hausdorff.In contrast, non-preregular spaces are encountered much more frequently in abstract algebra and algebraic geometry,in particular as the Zariski topology on an algebraic variety or the spectrum of a ring. They also arise in the modeltheory of intuitionistic logic: every complete Heyting algebra is the algebra of open sets of some topological space,but this space need not be preregular, much less Hausdorff.While the existence of unique limits for convergent nets and filters implies that a space is Hausdorff, there are non-Hausdorff T1 spaces in which every convergent sequence has a unique limit.[4]

27.4 Properties

Subspaces and products of Hausdorff spaces are Hausdorff,[5] but quotient spaces of Hausdorff spaces need not beHausdorff. In fact, every topological space can be realized as the quotient of some Hausdorff space.[6]

Hausdorff spaces are T1, meaning that all singletons are closed. Similarly, preregular spaces are R0.Another nice property of Hausdorff spaces is that compact sets are always closed.[7] This may fail in non-Hausdorffspaces such as Sierpiński space.The definition of a Hausdorff space says that points can be separated by neighborhoods. It turns out that this impliessomething which is seemingly stronger: in a Hausdorff space every pair of disjoint compact sets can also be separatedby neighborhoods,[8] in other words there is a neighborhood of one set and a neighborhood of the other, such that thetwo neighborhoods are disjoint. This is an example of the general rule that compact sets often behave like points.Compactness conditions together with preregularity often imply stronger separation axioms. For example, any locallycompact preregular space is completely regular. Compact preregular spaces are normal, meaning that they satisfyUrysohn’s lemma and the Tietze extension theorem and have partitions of unity subordinate to locally finite opencovers. The Hausdorff versions of these statements are: every locally compact Hausdorff space is Tychonoff, andevery compact Hausdorff space is normal Hausdorff.The following results are some technical properties regarding maps (continuous and otherwise) to and fromHausdorffspaces.

66 CHAPTER 27. HAUSDORFF SPACE

Let f : X → Y be a continuous function and suppose Y is Hausdorff. Then the graph of f, (x, f(x)) | x ∈ X , isa closed subset of X × Y.Let f : X → Y be a function and let ker(f) ≜ (x, x′) | f(x) = f(x′) be its kernel regarded as a subspace of X ×X.

• If f is continuous and Y is Hausdorff then ker(f) is closed.

• If f is an open surjection and ker(f) is closed then Y is Hausdorff.

• If f is a continuous, open surjection (i.e. an open quotient map) then Y is Hausdorff if and only if ker(f) isclosed.

If f,g : X→ Y are continuous maps and Y is Hausdorff then the equalizer eq(f, g) = x | f(x) = g(x) is closed inX. It follows that if Y is Hausdorff and f and g agree on a dense subset of X then f = g. In other words, continuousfunctions into Hausdorff spaces are determined by their values on dense subsets.Let f : X → Y be a closed surjection such that f−1(y) is compact for all y ∈ Y. Then if X is Hausdorff so is Y.Let f : X → Y be a quotient map with X a compact Hausdorff space. Then the following are equivalent

• Y is Hausdorff

• f is a closed map

• ker(f) is closed

27.5 Preregularity versus regularity

All regular spaces are preregular, as are all Hausdorff spaces. There are many results for topological spaces that holdfor both regular and Hausdorff spaces. Most of the time, these results hold for all preregular spaces; they were listedfor regular and Hausdorff spaces separately because the idea of preregular spaces came later. On the other hand,those results that are truly about regularity generally don't also apply to nonregular Hausdorff spaces.There are many situations where another condition of topological spaces (such as paracompactness or local com-pactness) will imply regularity if preregularity is satisfied. Such conditions often come in two versions: a regularversion and a Hausdorff version. Although Hausdorff spaces aren't generally regular, a Hausdorff space that is also(say) locally compact will be regular, because any Hausdorff space is preregular. Thus from a certain point of view,it is really preregularity, rather than regularity, that matters in these situations. However, definitions are usually stillphrased in terms of regularity, since this condition is better known than preregularity.See History of the separation axioms for more on this issue.

27.6 Variants

The terms “Hausdorff”, “separated”, and “preregular” can also be applied to such variants on topological spacesas uniform spaces, Cauchy spaces, and convergence spaces. The characteristic that unites the concept in all of theseexamples is that limits of nets and filters (when they exist) are unique (for separated spaces) or unique up to topologicalindistinguishability (for preregular spaces).As it turns out, uniform spaces, and more generally Cauchy spaces, are always preregular, so the Hausdorff condi-tion in these cases reduces to the T0 condition. These are also the spaces in which completeness makes sense, andHausdorffness is a natural companion to completeness in these cases. Specifically, a space is complete if and only ifevery Cauchy net has at least one limit, while a space is Hausdorff if and only if every Cauchy net has at most onelimit (since only Cauchy nets can have limits in the first place).

27.7. ALGEBRA OF FUNCTIONS 67

27.7 Algebra of functions

The algebra of continuous (real or complex) functions on a compact Hausdorff space is a commutative C*-algebra,and conversely by the Banach–Stone theorem one can recover the topology of the space from the algebraic propertiesof its algebra of continuous functions. This leads to noncommutative geometry, where one considers noncommutativeC*-algebras as representing algebras of functions on a noncommutative space.

27.8 Academic humour• Hausdorff condition is illustrated by the pun that in Hausdorff spaces any two points can be “housed off” fromeach other by open sets.[9]

• In the Mathematics Institute of at the University of Bonn, in which Felix Hausdorff researched and lectured,there is a certain room designated the Hausdorff-Raum. This is a pun, as Raum means both room and spacein German.

27.9 See also• Quasitopological space

• Weak Hausdorff space

• Fixed-point space, a Hausdorff space X such that every continuous function f:X→X has a fixed point.

27.10 Notes[1] Willard, pp. 86–87.

[2] Willard, pp. 86–87.

[3] Bourbaki, p. 75.

[4] van Douwen, Eric K. (1993). “An anti-Hausdorff Fréchet space in which convergent sequences have unique limits”.Topology and its Applications 51 (2): 147–158. doi:10.1016/0166-8641(93)90147-6.

[5] Hausdorff property is hereditary at PlanetMath.org.

[6] Shimrat, M. (1956). “Decomposition spaces and separation properties”. Quart. J. Math. 2: 128–129.

[7] Proof of A compact set in a Hausdorff space is closed at PlanetMath.org.

[8] Willard, p. 124.

[9] Colin Adams and Robert Franzosa. Introduction to Topology: Pure and Applied. p. 42

27.11 References• Arkhangelskii, A.V., L.S. Pontryagin, General Topology I, (1990) Springer-Verlag, Berlin. ISBN 3-540-18178-4.

• Bourbaki; Elements of Mathematics: General Topology, Addison-Wesley (1966).

• Hazewinkel, Michiel, ed. (2001), “Hausdorff space”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

Chapter 28

Hawaiian earring

In mathematics, the Hawaiian earring H is the topological space defined by the union of circles in the Euclideanplane R2 with center (1/n, 0) and radius 1/n for n = 1, 2, 3, .... The space H is homeomorphic to the one-pointcompactification of the union of a countably infinite family of open intervals.

-1

0

1

0 1 2

The Hawaiian earring. Only the ten largest circles are shown.

68

28.1. FUNDAMENTAL GROUP 69

The Hawaiian earring can be given a complete metric and it is compact. It is path connected but not semilocallysimply connected.The Hawaiian earring looks very similar to the wedge sum of countably infinitely many circles; that is, the rose withinfinitely many petals, but those two spaces are not homeomorphic. The difference between their topologies is seenin the fact that, in the Hawaiian earring, every open neighborhood of the point of intersection of the circles containsall but finitely many of the circles. It is also seen in the fact that the wedge sum is not compact: the complement ofthe distinguished point is a union of open intervals; to those add a small open neighborhood of the distinguished pointto get an open cover with no finite subcover.

28.1 Fundamental group

The Hawaiian earring is not simply connected, since the loop parametrising any circle is not homotopic to a trivialloop. Thus, it has a nontrivial fundamental group G.The Hawaiian earring H has the free group of countably infinitely many generators as a proper subgroup of itsfundamental group. G contains additional elements, which arise from loops whose image is not contained in finitelymany of the Hawaiian earring’s circles; in fact, some of them are surjective. For example, the path that on the interval[2−n, 2−(n−1)] circumnavigates the nth circle.It has been shown that G embeds into the inverse limit of the free groups with n generators, Fn, where the bondingmap from Fn to Fn₋₁ simply kills the last generator of Fn. However G is not the complete inverse limit but rather thesubgroup in which each generator appears only finitely many times. An example of an element of the inverse limitthat is not an element of G is an infinite commutator.G is uncountable, and it is not a free group. While its abelianisation has no known simple description, G has a normalsubgroup N such that G/N ≈

∏∞i=0 Z , the direct product of infinitely many copies of the infinite cyclic group (the

Baer–Specker group). This is called the infinite abelianization or strong abelianization of the Hawaiian earring, sincethe subgroup N is generated by elements where each coordinate (thinking of the Hawaiian earring as a subgroup ofthe inverse limit) is a product of commutators. In a sense, N can be thought of as the closure of the commutatorsubgroup.

28.2 References• Cannon, J. W.; Conner, G. R. (2000), “The big fundamental group, big Hawaiian earrings, and the big freegroups”, Topology and its Applications 106 (3): 273–291, doi:10.1016/S0166-8641(99)00104-2.

• Conner, G.; Spencer, K. (2005), “Anomalous behavior of the Hawaiian earring group”, Journal of GroupTheory 8 (2): 223–227, doi:10.1515/jgth.2005.8.2.223.

• Eda, K. (2002), “The fundamental groups of one-dimensional wild spaces and the Hawaiian earring” (PDF),Proceedings of the American Mathematical Society 130 (5): 1515–1522, doi:10.1090/S0002-9939-01-06431-0.

• Eda, K.; Kawamura, K. (2000), “The singular homology of the Hawaiian earring”, Journal of the LondonMathematical Society 62 (1): 305–310, doi:10.1112/S0024610700001071.

• Fabel, P. (2005), “The topological Hawaiian earring group does not embed in the inverse limit of free groups”(PDF), Algebraic & Geometric Topology 5: 1585–1587, doi:10.2140/agt.2005.5.1585.

• Morgan, J. W.; Morrison, I. (1986), “A van Kampen theorem for weak joins”, Proceedings of the LondonMathematical Society 53 (3): 562–576, doi:10.1112/plms/s3-53.3.562.

Chapter 29

Hedgehog space

In mathematics, a hedgehog space is a topological space, consisting of a set of spines joined at a point.For any cardinal number K , the K -hedgehog space is formed by taking the disjoint union of K real unit intervalsidentified at the origin. Each unit interval is referred to as one of the hedgehog’s spines. A K -hedgehog space issometimes called a hedgehog space of spininess K .The hedgehog space is a metric space, when endowed with the hedgehog metric d(x, y) = |x− y| if x and y lie inthe same spine, and by d(x, y) = x + y if x and y lie in different spines. Although their disjoint union makes theorigins of the intervals distinct, the metric identifies them by assigning them 0 distance.Hedgehog spaces are examples of real trees.[1]

29.1 Paris metric

The metric on the plane in which the distance between any two points is their Euclidean distance when the two pointsbelong to a ray though the origin, and is otherwise the sum of the distances of the two points from the origin, issometimes called the Paris metric[1] because navigation in this metric resembles that in the radial street plan ofParis. The Paris metric, restricted to the unit disk, is a hedgehog space where K is the cardinality of the continuum.

29.2 Kowalsky’s theorem

Kowalsky’s theorem, named after Hans-Joachim Kowalsky,[2] states that any metric space of weight K can be rep-resented as a topological subspace of the product of countably manyK -hedgehog spaces.

29.3 See also

• Comb space

• Long line (topology)

• Rose (topology)

29.4 Notes

[1] Carlisle (2007).

[2] Kowalsky (1961); Swardson (1979).

70

29.5. REFERENCES 71

29.5 References• Arkhangelskii, A. V.; Pontryagin, L. S. (1990), General Topology I, Berlin: Springer-Verlag, ISBN 3-540-18178-4.

• Carlisle, Sylvia (2007), “Model Theory of Real Trees”, Graduate Student Conference in Logic, Univ. of Illinois,Chicago.

• Kowalsky, H. J. (1961), Topologische Räume, Basel-Stuttgart: Birkhäuser.

• Steen, L. A.; Seebach, J. A., Jr. (1970), Counterexamples in Topology, Holt, Rinehart and Winston.

• Swardson, M. A. (1979), “A short proof of Kowalsky’s hedgehog theorem”, Proc. Amer. Math. Soc. 75 (1):188, doi:10.1090/s0002-9939-1979-0529240-7.

Chapter 30

Hilbert cube

Not to be confused with Hilbert curve.

In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructiveexample of some ideas in topology. Furthermore, many interesting topological spaces can be embedded in the Hilbertcube; that is, can be viewed as subspaces of the Hilbert cube (see below).

30.1 Definition

The Hilbert cube is best defined as the topological product of the intervals [0, 1/n] for n = 1, 2, 3, 4, ... That is,it is a cuboid of countably infinite dimension, where the lengths of the edges in each orthogonal direction form thesequence 1/nn∈N .The Hilbert cube is homeomorphic to the product of countably infinitely many copies of the unit interval [0, 1]. Inother words, it is topologically indistinguishable from the unit cube of countably infinite dimension.If a point in the Hilbert cube is specified by a sequence an with 0 ≤ an ≤ 1/n , then a homeomorphism to theinfinite dimensional unit cube is given by h(a)n = n · an .

30.2 The Hilbert cube as a metric space

It is sometimes convenient to think of the Hilbert cube as a metric space, indeed as a specific subset of a separableHilbert space (i.e. a Hilbert space with a countably infinite Hilbert basis). For these purposes, it is best not to thinkof it as a product of copies of [0,1], but instead as

[0,1] × [0,1/2] × [0,1/3] × ···;

as stated above, for topological properties, this makes no difference. That is, an element of the Hilbert cube is aninfinite sequence

(xn)

that satisfies

0 ≤ xn ≤ 1/n.

Any such sequence belongs to the Hilbert space ℓ2, so the Hilbert cube inherits a metric from there. One can showthat the topology induced by the metric is the same as the product topology in the above definition.

72

30.3. PROPERTIES 73

30.3 Properties

As a product of compact Hausdorff spaces, the Hilbert cube is itself a compact Hausdorff space as a result of theTychonoff theorem. The compactness of the Hilbert cube can also be proved without the Axiom of Choice byconstructing a continuous function from the usual Cantor set onto the Hilbert cube.In ℓ2, no point has a compact neighbourhood (thus, ℓ2 is not locally compact). One might expect that all of thecompact subsets of ℓ2 are finite-dimensional. The Hilbert cube shows that this is not the case. But the Hilbert cubefails to be a neighbourhood of any point p because its side becomes smaller and smaller in each dimension, so that anopen ball around p of any fixed radius e > 0 must go outside the cube in some dimension.Every subset of the Hilbert cube inherits from the Hilbert cube the properties of being both metrizable (and thereforeT4) and second countable. It is more interesting that the converse also holds: Every second countable T4 space ishomeomorphic to a subset of the Hilbert cube.Every Gδ-subset of the Hilbert cube is a Polish space, a topological space homeomorphic to a separable and completemetric space. Conversely, every Polish space is homeomorphic to a Gδ-subset of the Hilbert cube.[1]

30.4 Notes[1] Srivastava, pp. 55

30.5 References• Srivastava, Sashi Mohan (1998). A Course on Borel Sets. Graduate Texts in Mathematics. Springer-Verlag.ISBN 978-0-387-98412-4. Retrieved 12-04-08. Check date values in: |access-date= (help)

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978]. Counterexamples in Topology (Dover reprint of1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 507446.

Chapter 31

Hjalmar Ekdal topology

In mathematics, the Hjalmar Ekdal topology is a special example in the theory of topological spaces.[1]

The Hjalmar Ekdal topology consists of N* (the set of positive integers) together with the collection of all subsets ofN* in which every odd member is accompanied by its even successor. Examples: 2, 6, 9, 10If all such subsets are declared “open”, the “closed” subsets are consequently those in which every even member isaccompanied by its odd predecessor.It is not compact, but it is locally compact, paracompact and second countable.

31.1 Name

As it was the only original example (#55) in Steen and Seebach’s Counterexamples in Topology, it was named by theundergraduates who worked on it. According to John Feroe, now at Vassar College:

Since this was a group project among three professors and five students, we played with the idea ofchoosing a pseudonym as the author of the book. So the question was, if we were going to be someone,who should we be? I had just taken a course on Henrik Ibsen (this was, after all, at St Olaf College, aMinnesota college founded by Norwegian-American Lutherans and very true to its heritage which wasmy heritage as well for that matter). I had been particularly taken by the play The Wild Duck, whosemain character is a man named Hjalmar Ekdal. Hjalmar is a pathetic fellow who is unaware that almosteverything he has was provided for him—house, business, wife, even his child. He is also unaware thathe is quite incapable of succeeding on his own.

So we decided to call ourselves Hjalmar Ekdal since one way to look at what we were doing was collectingthe work and examples provided by others cataloging rather than creating. We put up a big sign in thelibrary alcove where we worked reading, “This space reserved for Hjalmar Ekdal,” and posted quotationsfrom Hjalmar Ekdal, such as “I haven¹t quite solved it yet, but I¹m working on it constantly.”

And although the resulting book carries the names of the supervising faculty as the authors, Hjalmardoes live on in that during that summer we had formulated a new example, and as its creators had theright to name it the Hjalmar Ekdal Topology ironically enough the only original example in the book.[2]

31.2 References[1] Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978]. Counterexamples in Topology (Dover reprint of 1978 ed.).

Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 507446.

[2] John Feroe (2002-04-21). “Hjalmar Ekdal Topology”. Newsgroup: sci.math.

74

Chapter 32

Homology sphere

In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for someinteger n ≥ 1. That is,

H0(X,Z) = Z = Hn(X,Z)

and

Hi(X,Z) = 0 for all other i.

Therefore X is a connected space, with one non-zero higher Betti number: bn. It does not follow that X is simplyconnected, only that its fundamental group is perfect (see Hurewicz theorem).A rational homology sphere is defined similarly but using homology with rational coefficients.

32.1 Poincaré homology sphere

The Poincaré homology sphere (also known as Poincaré dodecahedral space) is a particular example of a homologysphere. Being a spherical 3-manifold, it is the only homology 3-sphere (besides the 3-sphere itself) with a finitefundamental group. Its fundamental group is known as the binary icosahedral group and has order 120. This showsthe Poincaré conjecture cannot be stated in homology terms alone.

32.1.1 Construction

A simple construction of this space begins with a dodecahedron. Each face of the dodecahedron is identified withits opposite face, using the minimal clockwise twist to line up the faces. Gluing each pair of opposite faces togetherusing this identification yields a closed 3-manifold. (See Seifert–Weber space for a similar construction, using more“twist”, that results in a hyperbolic 3-manifold.)Alternatively, the Poincaré homology sphere can be constructed as the quotient space SO(3)/I where I is the icosahedralgroup (i.e. the rotational symmetry group of the regular icosahedron and dodecahedron, isomorphic to the alternatinggroup A5). More intuitively, this means that the Poincaré homology sphere is the space of all geometrically distin-guishable positions of an icosahedron (with fixed center and diameter) in Euclidean 3-space. One can also passinstead to the universal cover of SO(3) which can be realized as the group of unit quaternions and is homeomorphicto the 3-sphere. In this case, the Poincaré homology sphere is isomorphic to S3/Ĩ where Ĩ is the binary icosahedralgroup, the perfect double cover of I embedded in S3.Another approach is by Dehn surgery. The Poincaré homology sphere results from +1 surgery on the right-handedtrefoil knot.

75

76 CHAPTER 32. HOMOLOGY SPHERE

32.1.2 Cosmology

In 2003, lack of structure on the largest scales (above 60 degrees) in the cosmic microwave background as observedfor one year by the WMAP spacecraft led to the suggestion, by Jean-Pierre Luminet of the Observatoire de Paris andcolleagues, that the shape of the universe is a Poincaré sphere.[1][2] In 2008, astronomers found the best orientationon the sky for the model and confirmed some of the predictions of the model, using three years of observations bythe WMAP spacecraft.[3] However, there is no strong support for the correctness of the model, as yet.

32.2 Constructions and examples

• Surgery on a knot in the 3-sphere S3 with framing +1 or − 1 gives a homology sphere.

• More generally, surgery on a link gives a homology sphere whenever the matrix given by intersection numbers(off the diagonal) and framings (on the diagonal) has determinant +1 or −1.

• If p, q, and r are pairwise relatively prime positive integers then the link of the singularity xp + yq + zr = 0 (inother words, the intersection of a small 5-sphere around 0 with this complex surface) is a homology 3-sphere,called a Brieskorn 3-sphere Σ(p, q, r). It is homeomorphic to the standard 3-sphere if one of p, q, and r is 1,and Σ(2, 3, 5) is the Poincaré sphere.

• The connected sum of two oriented homology 3-spheres is a homology 3-sphere. A homology 3-sphere thatcannot be written as a connected sum of two homology 3-spheres is called irreducible or prime, and everyhomology 3-sphere can be written as a connected sum of prime homology 3-spheres in an essentially uniqueway. (See Prime decomposition (3-manifold).)

• Suppose that a1, ..., ar are integers all at least 2 such that any two are coprime. Then the Seifert fiber space

b, (o1, 0); (a1, b1), . . . , (ar, br)

over the sphere with exceptional fibers of degrees a1, ..., ar is a homology sphere, where the b's arechosen so that

b+ b1/a1 + · · ·+ br/ar = 1/(a1 · · · ar).

(There is always a way to choose the b′s, and the homology sphere does not depend (up to isomorphism)on the choice of b′s.) If r is at most 2 this is just the usual 3-sphere; otherwise they are distinct non-trivialhomology spheres. If the a′s are 2, 3, and 5 this gives the Poincaré sphere. If there are at least 3 a′s, not2, 3, 5, then this is an acyclic homology 3-sphere with infinite fundamental group that has a Thurstongeometry modeled on the universal cover of SL2(R).

32.3 Invariants

• The Rokhlin invariant is a Z/2Z valued invariant of homology 3-spheres.

• The Casson invariant is an integer valued invariant of homology 3-spheres, whose reduction mod 2 is theRokhlin invariant.

32.4. APPLICATIONS 77

32.4 Applications

If A is a homology 3-sphere not homeomorphic to the standard 3-sphere, then the suspension of A is an example of a4-dimensional homology manifold that is not a topological manifold. The double suspension of A is homeomorphicto the standard 5-sphere, but its triangulation (induced by some triangulation of A) is not a PL manifold. In otherwords, this gives an example of a finite simplicial complex that is a topological manifold but not a PL manifold. (Itis not a PL manifold because the link of a point is not always a 4-sphere.)Galewski and Stern showed that all compact topological manifolds (without boundary) of dimension at least 5 arehomeomorphic to simplicial complexes if and only if there is a homology 3 sphere Σ with Rokhlin invariant 1 suchthat the connected sum Σ#Σ of Σ with itself bounds a smooth acyclic 4-manifold. As of 2013 the existence of sucha homology 3-sphere was an unsolved problem. On March 11, 2013, Ciprian Manolescu posted a preprint on theArXiv[4] claiming to show that there is no such homology sphere with the given property, and therefore, there are5-manifolds not homeomorphic to simplicial complexes. In particular, the example originally given by Galewskiand Stern (see Galewski and Stern, A universal 5-manifold with respect to simplicial triangulations, in GeometricTopology (Proceedings Georgia Topology Conference, Athens Georgia, 1977, Academic Press, New York, pp 345–350)) is not triangulable.

32.5 References[1] “Is the universe a dodecahedron?", article at PhysicsWorld.

[2] Luminet, Jean-Pierre; Jeff Weeks, Alain Riazuelo, Roland Lehoucq, Jean-Phillipe Uzan (2003-10-09). “Dodecahedralspace topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background”.Nature 425 (6958): 593–595. arXiv:astro-ph/0310253. Bibcode:2003Natur.425..593L. doi:10.1038/nature01944. PMID14534579.

[3] Roukema, Boudewijn; Zbigniew Buliński; Agnieszka Szaniewska; Nicolas E. Gaudin (2008). “A test of the Poincaredodecahedral space topology hypothesis with the WMAP CMB data”. Astronomy and Astrophysics 482 (3): 747–753.arXiv:0801.0006. Bibcode:2008A&A...482..747L. doi:10.1051/0004-6361:20078777.

[4] Manolescu, Ciprian. “Pin(2)-equivariant Seiberg-Witten Floer homology and the TriangulationConjecture”. arXiv:1303.2354.To appear in Journal of the AMS.

32.6 Selected reading• Emmanuel Dror, Homology spheres, Israel Journal of Mathematics 15 (1973), 115–129. MR 0328926

• David Galewski, Ronald Stern Classification of simplicial triangulations of topological manifolds, Annals ofMathematics 111 (1980), no. 1, pp. 1–34.

• Robion Kirby, Martin Scharlemann, Eight faces of the Poincaré homology 3-sphere. Geometric topology (Proc.Georgia Topology Conf., Athens, Ga., 1977), pp. 113–146, Academic Press, New York-London, 1979.

• Michel Kervaire, Smooth homology spheres and their fundamental groups, Transactions of the AmericanMath-ematical Society 144 (1969) 67–72. MR 0253347

• Nikolai Saveliev, Invariants of Homology 3-Spheres, Encyclopaedia of Mathematical Sciences, vol 140. Low-Dimensional Topology, I. Springer-Verlag, Berlin, 2002. MR 1941324 ISBN 3-540-43796-7

32.7 External links• A 16-Vertex Triangulation of the Poincaré Homology 3-Sphere and Non-PL Spheres with Few Vertices byAnders Björner and Frank H. Lutz

• Lecture by David Gillman on The best picture of Poincare’s homology sphere

• A cosmic hall of mirrors - physicsworld (26 Sep 2005)

Chapter 33

Homotopy sphere

In algebraic topology, a branch of mathematics, a homotopy sphere is an n-manifold that is homotopy equivalentto the n-sphere. It thus has the same homotopy groups and the same homology groups as the n-sphere, and so everyhomotopy sphere is necessarily a homology sphere.The topological generalized Poincaré conjecture is that any n-dimensional homotopy sphere is homeomorphic to then-sphere; it was solved by Stephen Smale in dimensions five and higher, by Michael Freedman in dimension 4, andfor dimension 3 by Grigori Perelman in 2005.The resolution of the smooth Poincaré conjecture in dimensions 5 and larger implies that homotopy spheres in thosedimensions are precisely exotic spheres. It is still an open question (as of 2014) whether or not there are non-trivialsmooth homotopy spheres in dimension 4.

33.1 References• A. Kosinski, Differential Manifolds. Academic Press 1993.

33.2 See also• Homology sphere

• Homotopy groups of spheres

• Poincaré conjecture

78

Chapter 34

Hyperbolic space

A perspective projection of a dodecahedral tessellation in H3.Four dodecahedra meet at each edge, and eight meet at each vertex, like the cubes of a cubic tessellation in E3

In mathematics, hyperbolic space is a homogeneous space that has a constant negative curvature, where in this casethe curvature is the sectional curvature. It is hyperbolic geometry in more than 2 dimensions, and is distinguished

79

80 CHAPTER 34. HYPERBOLIC SPACE

from Euclidean spaces with zero curvature that define the Euclidean geometry, and elliptic geometry that have aconstant positive curvature.When embedded to a Euclidean space (of a higher dimension), every point of a hyperbolic space is a saddle point. An-other distinctive property is the amount of space covered by the n-ball in hyperbolic n-space: it increases exponentiallywith respect to the radius of the ball for large radii, rather than polynomially.

34.1 Formal definition

Hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian man-ifold with a constant negative sectional curvature. Hyperbolic space is a space exhibiting hyperbolic geometry. It isthe negative-curvature analogue of the n-sphere. Although hyperbolic space Hn is diffeomorphic to Rn, its negative-curvature metric gives it very different geometric properties.Hyperbolic 2-space, H2, is also called the hyperbolic plane.

34.2 Models of hyperbolic space

Hyperbolic space, developed independently by Nikolai Lobachevsky and János Bolyai, is a geometrical space analo-gous to Euclidean space, but such that Euclid’s parallel postulate is no longer assumed to hold. Instead, the parallelpostulate is replaced by the following alternative (in two dimensions):

• Given any line L and point P not on L, there are at least two distinct lines passing through P which do notintersect L.

It is then a theorem that there are infinitely many such lines through P. This axiom still does not uniquely characterizethe hyperbolic plane up to isometry; there is an extra constant, the curvatureK < 0, whichmust be specified. However,it does uniquely characterize it up to homothety, meaning up to bijections which only change the notion of distanceby an overall constant. By choosing an appropriate length scale, one can thus assume, without loss of generality, thatK = −1.Models of hyperbolic spaces that can be embedded in a flat (e.g. Euclidean) spaces may be constructed. In particular,the existence of model spaces implies that the parallel postulate is logically independent of the other axioms ofEuclidean geometry.There are several important models of hyperbolic space: the Klein model, the hyperboloid model, the Poincaréball model and the Poincaré half space model. These all model the same geometry in the sense that any two ofthem can be related by a transformation that preserves all the geometrical properties of the space, including isometry(though not with respect to the metric of a Euclidean embedding).

34.2.1 The hyperboloid model

Main article: Hyperboloid model

The hyperboloid model realizes hyperbolic space as a hyperboloid in Rn+1 = (x0,...,xn)|xi∈R, i=0,1,...,n. The hy-perboloid is the locus Hn of points whose coordinates satisfy

x20 − x2

1 − · · · − x2n = 1, x0 > 0.

In this model a line (or geodesic) is the curve formed by the intersection of Hn with a plane through the origin inRn+1.The hyperboloid model is closely related to the geometry of Minkowski space. The quadratic form

Q(x) = x20 − x2

1 − x22 − · · · − x2

n,

34.2. MODELS OF HYPERBOLIC SPACE 81

which defines the hyperboloid, polarizes to give the bilinear form

B(x, y) = (Q(x+ y)−Q(x)−Q(y))/2 = x0y0 − x1y1 − · · · − xnyn.

The space Rn+1, equipped with the bilinear form B, is an (n+1)-dimensional Minkowski space Rn,1.One can associate a distance on the hyperboloid model by defining[1] the distance between two points x and y on Hto be

d(x, y) = arcoshB(x, y).

This function satisfies the axioms of a metric space. It is preserved by the action of the Lorentz group on Rn,1. Hencethe Lorentz group acts as a transformation group preserving isometry on Hn.

34.2.2 The Klein model

Main article: Klein model

An alternative model of hyperbolic geometry is on a certain domain in projective space. The Minkowski quadraticform Q defines a subset Un ⊂ RPn given as the locus of points for which Q(x) > 0 in the homogeneous coordinates x.The domain Un is the Klein model of hyperbolic space.The lines of this model are the open line segments of the ambient projective space which lie in Un. The distancebetween two points x and y in Un is defined by

d(x, y) = arcosh(

B(x, y)√Q(x)Q(y)

).

This is well-defined on projective space, since the ratio under the inverse hyperbolic cosine is homogeneous of degree0.This model is related to the hyperboloid model as follows. Each point x ∈ Un corresponds to a line Lx throughthe origin in Rn+1, by the definition of projective space. This line intersects the hyperboloid Hn in a unique point.Conversely, through any point on Hn, there passes a unique line through the origin (which is a point in the projectivespace). This correspondence defines a bijection between Un and Hn. It is an isometry, since evaluating d(x,y) alongQ(x) = Q(y) = 1 reproduces the definition of the distance given for the hyperboloid model.

34.2.3 The Poincaré ball model

Main article: Poincaré disc model

A closely related pair of models of hyperbolic geometry are the Poincaré ball and Poincaré half-space models.The ball model comes from a stereographic projection of the hyperboloid in Rn+1 onto the hyperplane x0 = 0. Indetail, let S be the point in Rn,1 with coordinates (−1,0,0,...,0): the South pole for the stereographic projection. Foreach point P on the hyperboloid Hn, let P∗ be the unique point of intersection of the line SP with the plane x0 = 0.This establishes a bijective mapping of Hn into the unit ball

Bn = (x1, . . . , xn)|x21 + · · ·+ x2

n < 1

in the plane x0 = 0.The geodesics in this model are semicircles that are perpendicular to the boundary sphere of Bn. Isometries of theball are generated by spherical inversion in hyperspheres perpendicular to the boundary.

82 CHAPTER 34. HYPERBOLIC SPACE

34.2.4 The Poincaré half space model

Main article: Poincaré half-plane model

The half-space model results from applying inversion in a circle with centre a boundary point of the Poincaré ballmodel Bn above and a radius of twice the radius.This sends circles to circles and lines, and is moreover a conformal transformation. Consequently, the geodesics ofthe half-space model are lines and circles perpendicular to the boundary hyperplane.

34.3 Hyperbolic manifolds

Every complete, connected, simply connected manifold of constant negative curvature −1 is isometric to the realhyperbolic space Hn. As a result, the universal cover of any closed manifold M of constant negative curvature −1,which is to say, a hyperbolic manifold, is Hn. Thus, every such M can be written as Hn/Γ where Γ is a torsion-freediscrete group of isometries on Hn. That is, Γ is a lattice in SO+(n,1).

34.3.1 Riemann surfaces

Two-dimensional hyperbolic surfaces can also be understood according to the language of Riemann surfaces. Accord-ing to the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. Most hyperbolicsurfaces have a non-trivial fundamental group π1=Γ; the groups that arise this way are known as Fuchsian groups.The quotient space H²/Γ of the upper half-plane modulo the fundamental group is known as the Fuchsian model ofthe hyperbolic surface. The Poincaré half plane is also hyperbolic, but is simply connected and noncompact. It is theuniversal cover of the other hyperbolic surfaces.The analogous construction for three-dimensional hyperbolic surfaces is the Kleinian model.

34.4 See also• Mostow rigidity theorem• Hyperbolic manifold• Hyperbolic 3-manifold• Murakami–Yano formula• Pseudosphere• Dini’s surface

34.5 References[1] Note the similarity with the chordal metric on a sphere, which uses trigonometric instead of hyperbolic functions.

• A'Campo, Norbert and Papadopoulos, Athanase, (2012) Notes on hyperbolic geometry, in: Strasbourg Masterclass on Geometry, pp. 1–182, IRMA Lectures in Mathematics and Theoretical Physics, Vol. 18, Zürich:European Mathematical Society (EMS), 461 pages, SBN ISBN 978-3-03719-105-7, DOI 10.4171/105.

• Ratcliffe, John G., Foundations of hyperbolic manifolds, New York, Berlin. Springer-Verlag, 1994.• Reynolds, William F. (1993) “Hyperbolic Geometry on a Hyperboloid”, American Mathematical Monthly100:442–455.

• Wolf, Joseph A. Spaces of constant curvature, 1967. See page 67.• Hyperbolic Voronoi diagrams made easy, Frank Nielsen

Chapter 35

Infinite broom

y1

0.5

0.5 1

x

Standard infinite broom

In topology, the infinite broom is a subset of the Euclidean plane that is used as an example distinguishing variousnotions of connectedness. The closed infinite broom is the closure of the infinite broom, and is also referred to asthe broom space.[1]

83

84 CHAPTER 35. INFINITE BROOM

35.1 Definition

The infinite broom is the subset of the Euclidean plane that consists of all closed line segments joining the origin tothe point (1, 1 / n) as n varies over all positive integers, together with the interval (½, 1] on the x-axis.[2]

The closed infinite broom is then the infinite broom together with the interval (0, ½] on the x-axis. In other words, itconsists of all closed line segments joining the origin to the point (1, 1 / n) or to the point (1, 0).[2]

35.2 Properties

Both the infinite broom and its closure are connected, as every open set in the plane which contains the segment onthe x-axis must intersect slanted segments. Neither are locally connected. Despite the closed infinite broom beingarc connected, the standard infinite broom is not path connected.[2]

35.3 See also• Comb space

• Integer broom topology

35.4 References[1] Chapter 6 exercise 3.5 of Joshi, K. D. (1983), Introduction to general topology, New York: John Wiley & Sons, Inc., ISBN

978-0-85226-444-7, MR 0709260

[2] Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.),Mineola, NY: Dover Publications, Inc., p. 139, ISBN 978-0-486-68735-3, MR 1382863

Chapter 36

Infinite loop space machine

In topology, a branch of mathematics, given a topological monoid X up to homotopy (in a nice way), an infinite loopspace machine produces a group completion of X together with infinite loop space structure. For example, one cantake X to be the classifying space of a symmetric monoidal category S; that is,X = BS . Then the machine producesthe group completion BS → K(S) . The spaceK(S) may be described by the K-theory spectrum of S.

36.1 References• May, The uniqueness of infinite loop space machine

85

Chapter 37

Interlocking interval topology

Not to be confused with Overlapping interval topology.

In mathematics, and especially general topology, the interlocking interval topology is an example of a topology onthe set S := R+ \ Z+, i.e. the set of all positive real numbers that are not positive whole numbers.[1] To give the set Sa topology means to say which subsets of S are “open”, and to do so in a way that the following axioms are met:[2]

1. The union of open sets is an open set.

2. The finite intersection of open sets is an open set.

3. S and the empty set ∅ are open sets.

37.1 Construction

The open sets in this topology are taken to be the whole set S, the empty set ∅, and the sets generated by

Xn :=

(0,

1

n

)∪ (n, n+ 1) =

x ∈ R+ : 0 < x <

1

nor n < x < n+ 1

.

The sets generated by Xn will be formed by all possible unions of finite intersections of the Xn.[3]

37.2 References[1] Steen & Seebach (1978) pp.77 – 78

[2] Steen & Seebach (1978) p.3

[3] Steen & Seebach (1978) p.4

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1978). Counterexamples in Topology (2nd ed.). Berlin, New York:Springer-Verlag. ISBN 3-540-90312-7. MR 507446. Zbl 0386.54001.

86

Chapter 38

Irrational winding of a torus

In topology, a branch of mathematics, an irrational winding of a torus is a continuous injection of a line into a torusthat is used to set up several counterexamples.[1] A related notion is the Kronecker foliation of a torus, a foliationformed by the set of all translates of a given irrational winding.

38.1 Definition

One way of constructing a torus is as the quotient space T 2 = R2/Z2 of a two-dimensional real vector space by theadditive subgroup of integer vectors, with the corresponding projection π : R2 → T 2 . Each point in the torus has asits preimage one of the translates of the square lattice Z2 in R2 , and π factors through a map that takes any point inthe plane to a point in the unit square [0, 1)2 given by the fractional parts of the original point’s Cartesian coordinates.Now consider a line inR2 given by the equation y = kx. If the slope k of the line is rational, then it can be representedby a fraction and a corresponding lattice point of Z2 . It can be shown that then the projection of this line is a simpleclosed curve on a torus. If, however, k is irrational, then it will not cross any lattice points except 0, which meansthat its projection on the torus will not be a closed curve, and the restriction of π on this line is injective. Moreover,it can be shown that the image of this restricted projection as a subspace, called the irrational winding of a torus, isdense in the torus.

38.2 Applications

Irrational windings of a torus may be used to set up counter-examples related to monomorphisms. An irrationalwinding is an immersed submanifold but not a regular submanifold of the torus, which shows that the image of amanifold under a continuous injection to another manifold is not necessarily a (regular) submanifold.[2] Irrationalwindings are also examples of the fact that the induced submanifold topology does not have to coincide with thesubspace topology of the submanifold [2] a[›]

Secondly, the torus can be considered as a Lie group U(1) × U(1) , and the line can be considered as R . Thenit is easy to show that the image of the continuous and analytic group homomorphism x 7→ (eix, eikx) is not a Liesubgroup[2][3] (because it’s not closed in the torus – see the closed subgroup theorem) while, of course, it is still agroup. It may also be used to show that if a subgroup H of the Lie group G is not closed, the quotient G/H does notneed to be a submanifold[4] and might even fail to be a Hausdorff space.

38.3 See also

• Torus knot

87

88 CHAPTER 38. IRRATIONAL WINDING OF A TORUS

38.4 Notes

^ a: As a topological subspace of the torus, the irrational winding is not a manifold at all, because it is not locallyhomeomorphic to R

38.5 References[1] D. P. Zhelobenko. Compact Lie groups and their representations.

[2] Loring W. Tu (2010). An Introduction to Manifolds. Springer. p. 168. ISBN 978-1-4419-7399-3.

[3] Čap, Andreas; Slovák, Jan (2009), Parabolic Geometries: Background and general theory, AMS, p. 24, ISBN 978-0-8218-2681-2

[4] Sharpe, R.W. (1997), Differential Geometry: Cartan’s Generalization of Klein’s Erlangen Program, Springer-Verlag, NewYork, p. 146, ISBN 0-387-94732-9

Chapter 39

K-topology

In mathematics, particularly topology, theK-topology is a topology that one can impose on the set of all real numberswhich has some interesting properties. Relative to the set of all real numbers carrying the standard topology, the setK = 1/n | n is a natural number is not closed since it doesn’t contain its (only) limit point 0. Relative to the K-topology however, the set K is automatically decreed to be closed by adding ‘more’ basis elements to the standardtopology on R. Basically, the K-topology on R is strictly finer than the standard topology on R. It is mostly useful forcounterexamples in basic topology.

39.1 Formal definition

Let R be the set of all real numbers and let K = 1/n | n is a natural number. Generate a topology on R by takingas basis all open intervals (a, b) and all sets of the form (a, b) – K (the set of all elements in (a, b) that are not in K).The topology generated is known as the K-topology on R.Note that: The sets described in the definition do form a basis (they satisfy the conditions to be a basis).

39.2 Properties and examples

Throughout this section, T will denote the K-topology and (R, T) will denote the set of all real numbers with theK-topology as a topological space.1. The topology T on R is strictly finer than the standard topology on R but not comparable with the lower limittopology on R2. From the previous example, it follows that (R, T) is not compact3. (R, T) is Hausdorff but not regular. The fact that it is Hausdorff follows from the first property. It is not regularsince the closed set K and the point 0 have no disjoint neighbourhoods about them4. Surprisingly enough, (R, T) is a connected topological space. However, (R, T) is not path connected; it has preciselytwo path components: (−∞, 0] and (0, +∞)5. Note also that (R, T) is not locally path connected (since its path components are not equal to its components). Itis also not locally connected at 0 but it is locally connected everywhere else6. The closed interval [0,1] is not compact as a subspace of (R, T) since it is not even limit point compact (K is aninfinite subspace of [0,1] that has no limit point in [0,1])7. In fact, no subspace of (R, T) containing K can be compact. If A were a subspace of (R, T) containing K, K wouldhave no limit point in A so that A can not be limit point compact. Therefore, A cannot be compact8. The quotient space of (R, T) obtained by collapsing K to a point is not Hausdorff. K is distinct from 0, but can'tbe separated from 0 by disjoint open sets.

89

90 CHAPTER 39. K-TOPOLOGY

39.3 See also• Lower limit topology

• Natural topology

• Sequence

• Locally connected space

• Connected space

39.4 References• James Munkres (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.

Chapter 40

Knaster–Kuratowski fan

x

y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

The Knaster-Kuratowski fan, or "Cantor’s teepee"

In topology, a branch ofmathematics, theKnaster–Kuratowski fan (also known asCantor’s leaky tent orCantor’steepee depending on the presence or absence of the apex) is a connected topological space with the property that theremoval of a single point makes it totally disconnected.LetC be the Cantor set, let p be the point ( 12 , 1

2 ) ∈ R2 , and let L(c) , for c ∈ C , denote the line segment connecting(c, 0) to p . If c ∈ C is an endpoint of an interval deleted in the Cantor set, let Xc = (x, y) ∈ L(c) : y ∈ Q ;for all other points in C let Xc = (x, y) ∈ L(c) : y /∈ Q ; the Knaster–Kuratowski fan is defined as

∪c∈C Xc

equipped with the subspace topology inherited from the standard topology on R2 .The fan itself is connected, but becomes totally disconnected upon the removal of p .

91

92 CHAPTER 40. KNASTER–KURATOWSKI FAN

40.1 References• Knaster, B.; Kuratowski, C. (1921), “Sur les ensembles connexes” (PDF), Fundamenta Mathematicae 2 (1):206–255

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446

40.2 External links

Chapter 41

Lexicographic order topology on the unitsquare

In general topology, the lexicographic ordering on the unit square is a topology on the unit square S, i.e. on theset of points (x,y) in the plane such that 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1.[1]

41.1 Construction

As the name suggests, we use the lexicographical ordering on the square to define a topology. Given two points inthe square, say (x,y) and (u,v), we say that (x,y) ≺ (u,v) if and only if either x < u or both x = u and y < v. Given thelexicographical ordering on the square, we use the order topology to define the topology on S.

41.2 Properties

The order topology makes S into a completely normal Hausdorff space. [1] It is an example of an order topology inwhich there are uncountably many pairwise-disjoint homeomorphic copies of the real line. Since the lexicographicalorder on S can be proven to be complete, then this topology makes S into a compact set. At the same time, S is notseparable, since the set of all points of the form (x,1/2) is discrete but is uncountable. Hence S is not metrizable(since any compact metric space is separable); however, it is first countable. [1]

41.3 See also• Long line

41.4 References[1] Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, p. 73, ISBN 0-486-68735-X

93

Chapter 42

List of examples in general topology

This is a list of useful examples in general topology, a field of mathematics.

• Alexandrov topology

• Cantor space

• Co-kappa topology

• Cocountable topology• Cofinite topology

• Compact-open topology

• Compactification

• Discrete topology

• Double-pointed cofinite topology

• Extended real number line

• Finite topological space

• Hawaiian earring

• Hilbert cube

• Irrational cable on a torus

• Lakes of Wada

• Long line

• Order topology

• Lexicographical/dictionary order• Ordinal number topology• Real line• Split interval

• Overlapping interval topology

• Moore plane

• Sierpiński space

• Sorgenfrey line

94

42.1. SEE ALSO 95

• Sorgenfrey plane

• Space-filling curve

• Topologist’s sine curve

• Trivial topology

• Unit interval

• Zariski topology

42.1 See also• Counterexamples in Topology

• π-Base: An Interactive Encyclopedia of Topological Spaces

Chapter 43

Long line (topology)

In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in acertain way “longer”. It behaves locally just like the real line, but has different large-scale properties (e.g., it is neitherLindelöf nor separable). Therefore it serves as one of the basic counterexamples of topology.[1] Intuitively, the usualreal-number line consists of a countable number of line segments [0, 1) laid end-to-end, whereas the long line isconstructed from an uncountable number of such segments.

43.1 Definition

The closed long ray L is defined as the cartesian product of the first uncountable ordinal ω1 with the half-openinterval [0, 1), equipped with the order topology that arises from the lexicographical order on ω1 × [0, 1). The openlong ray is obtained from the closed long ray by removing the smallest element (0,0).The long line is obtained by putting together a long ray in each direction. More rigorously, it can be defined as theorder topology on the disjoint union of the reversed open long ray (“reversed” means the order is reversed) and the(not reversed) closed long ray, totally ordered by letting the points of the latter be greater than the points of the former.Alternatively, take two copies of the open long ray and identify the open interval 0 × (0, 1) of the one with the sameinterval of the other but reversing the interval, that is, identify the point (0, t) (where t is a real number such that 0 <t < 1) of the one with the point (0,1 − t) of the other, and define the long line to be the topological space obtained bygluing the two open long rays along the open interval identified between the two. (The former construction is betterin the sense that it defines the order on the long line and shows that the topology is the order topology; the latter isbetter in the sense that it uses gluing along an open set, which is clearer from the topological point of view.)Intuitively, the closed long ray is like a real (closed) half-line, except that it is much longer in one direction: we saythat it is long at one end and closed at the other. The open long ray is like the real line (or equivalently an openhalf-line) except that it is much longer in one direction: we say that it is long at one end and short (open) at the other.The long line is longer than the real lines in both directions: we say that it is long in both directions.However, many authors speak of the “long line” where we have spoken of the (closed or open) long ray, and thereis much confusion between the various long spaces. In many uses or counterexamples, however, the distinction isunessential, because the important part is the “long” end of the line, and it doesn't matter what happens at the otherend (whether long, short, or closed).A related space, the (closed) extended long ray, L*, is obtained as the one-point compactification of L by adjoiningan additional element to the right end of L. One can similarly define the extended long line by adding two elementsto the long line, one at each end.

43.2 Properties

The closed long ray L = ω1 × [0,1) consists of an uncountable number of copies of [0,1) 'pasted together' end-to-end.Compare this with the fact that for any countable ordinal α, pasting together α copies of [0,1) gives a space which isstill homeomorphic (and order-isomorphic) to [0,1). (And if we tried to glue together more than ω1 copies of [0,1),

96

43.3. P-ADIC ANALOG 97

the resulting space would no longer be locally homeomorphic to R.)Every increasing sequence in L converges to a limit in L; this is a consequence of the facts that (1) the elements of ω1

are the countable ordinals, (2) the supremum of every countable family of countable ordinals is a countable ordinal,and (3) every increasing and bounded sequence of real numbers converges. Consequently, there can be no strictlyincreasing function L→R.As order topologies, the (possibly extended) long rays and lines are normal Hausdorff spaces. All of them have thesame cardinality as the real line, yet they are 'much longer'. All of them are locally compact. None of them ismetrisable; this can be seen as the long ray is sequentially compact but not compact, or even Lindelöf.The (non-extended) long line or ray is not paracompact. It is path-connected, locally path-connected and simplyconnected but not contractible. It is a one-dimensional topological manifold, with boundary in the case of the closedray. It is first-countable but not second countable and not separable, so authors who require the latter properties intheir manifolds do not call the long line a manifold.[2]

The long line or ray can be equipped with the structure of a (non-separable) differentiable manifold (with boundaryin the case of the closed ray). However, contrary to the topological structure which is unique (topologically, thereis only one way to make the real line “longer” at either end), the differentiable structure is not unique: in fact, foreach natural number k there exist infinitely many Ck+1 or C∞ structures on the long line or ray inducing any given Ck

structure on it.[3] This is in sharp contrast with the situation for ordinary (that is, separable) manifolds, where a Ck

structure uniquely determines a C∞ structure as soon as k≥1.It makes sense to consider all the long spaces at once because every connected (non-empty) one-dimensional (notnecessarily separable) topological manifold possibly with boundary, is homeomorphic to either the circle, the closedinterval, the open interval (real line), the half-open interval, the closed long ray, the open long ray, or the long line.[4]

The long line or ray can even be equipped with the structure of a (real) analytic manifold (with boundary in the caseof the closed ray). However, this is much more difficult than for the differentiable case (it depends on the classificationof (separable) one-dimensional analytic manifolds, which is more difficult than for differentiable manifolds). Again,any given C∞ structure can be extended in infinitely many ways to different Cω (=analytic) structures.[5]

The long line or ray cannot be equipped with a Riemannian metric that induces its topology. The reason is thatRiemannian manifolds, even without the assumption of paracompactness, can be shown to be metrizable.[6]

The extended long ray L* is compact. It is the one-point compactification of the closed long ray L, but it is also itsStone-Čech compactification, because any continuous function from the (closed or open) long ray to the real line iseventually constant.[7] L* is also connected, but not path-connected because the long line is 'too long' to be coveredby a path, which is a continuous image of an interval. L* is not a manifold and is not first countable.

43.3 p-adic analog

There exists a p-adic analog of the long line, which is due to George Bergman.[8]

This space is constructed as the increasing union of an uncountable directed set of copies Xᵧ of the ring of p-adicintegers, indexed by a countable ordinal γ. Define a map from Xδ to Xᵧ whenever δ<γ as follows:

• If γ is a successor ε+1 then the map from Xε to Xᵧ is just multiplication by p. For other δ the map from Xδ toXᵧ is the composition of the map from Xδ to Xε and the map from Xε to Xᵧ

• If γ is a limit ordinal then the direct limit of the sets Xδ for δ<γ is a countable union of p-adic balls, so can beembedded in Xᵧ, as Xᵧ with a point removed is also a countable union of p-adic balls. This defines compatibleembeddings of Xδ intoto Xᵧ for all δ<γ.

This space is not compact, but the union of any countable set of compact subspaces has compact closure.

43.4 Higher dimensions

Some examples of non-paracompact manifolds in higher dimensions include the Prüfer manifold, products of anynon-paracompact manifold with any non-empty manifold, the ball of long radius, and so on. The bagpipe theoremshows that there are 2ℵ1 isomorphism classes of non-paracompact surfaces.

98 CHAPTER 43. LONG LINE (TOPOLOGY)

There are no complex analogues of the long line as every Riemann surface is paracompact, but Calabi & Rosenlicht(1953) gave an example of a non-paracompact complex manifold of complex dimension 2.

43.5 See also• Lexicographic order topology on the unit square

43.6 References[1] Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978]. Counterexamples in Topology (Dover reprint of 1978 ed.).

Berlin, New York: Springer-Verlag. pp. 71–72. ISBN 978-0-486-68735-3. MR 507446. Zbl 1245.54001.

[2] Shastri, Anant R. (2011), Elements of Differential Topology, CRC Press, p. 122, ISBN 9781439831632.

[3] Koch, Winfried & Puppe, Dieter (1968). “Differenzierbare Strukturen auf Mannigfaltigkeiten ohne abzaehlbare Basis”.Archiv der Mathematik 19: 95–102. doi:10.1007/BF01898807.

[4] Kunen, K.; Vaughan, J. (2014), Handbook of Set-Theoretic Topology, Elsevier, p. 643, ISBN 9781483295152.

[5] Kneser, H. & Kneser, M. (1960). “Reell-analytische Strukturen der Alexandroff-Halbgeraden und der Alexandroff-Geraden”. Archiv der Mathematik 11: 104–106. doi:10.1007/BF01236917.

[6] S. Kobayashi and K. Nomizu (1963). Foundations of differential geometry I. Interscience. p. 166.

[7] Joshi, K. D. (1983). “Chapter 15 Section 3”. Introduction to general topology. Jon Wiley and Sons. ISBN 0-470-27556-1.MR 709260.

[8] Serre, Jean-Pierre. “IV (“Analytic Manifolds”), appendix 3 (“The Transfinite p-adic line”)". Lie Algebras and Lie Groups(1964 Lectures given at Harvard University). Lecture Notes in Mathematics part II (“Lie Groups”). Springer-Verlag. ISBN3-540-55008-9.

• Calabi, Eugenio; Rosenlicht, Maxwell (1953), “Complex analytic manifolds without countable base”, Proc.Amer. Math. Soc. 4: 335–340, doi:10.1090/s0002-9939-1953-0058293-x, MR 0058293

Chapter 44

Loop space

In topology, a branch of mathematics, the loop spaceΩX of a pointed topological space X is the space of based mapsfrom the circle S1 to X with the compact-open topology. Two elements of a loop space can be naturally concatenated.With this concatenation operation, a loop space is an A∞-space. The adjective A∞ describes the manner in whichconcatenating loops is homotopy coherently associative.The quotient of the loop space ΩX by the equivalence relation of pointed homotopy is the fundamental group π1(X).The iterated loop spaces of X are formed by applying Ω a number of times.An analogous construction of topological spaces without basepoint is the free loop space. The free loop space of atopological space X is the space of maps from S1 to X with the compact-open topology. That is to say, the free loopspace of a topological space X is the function space Map(S1, X) . The free loop space of X is denoted by LX .The free loop space construction is right adjoint to the cartesian product with the circle, while the loop space con-struction is right adjoint to the reduced suspension. This adjunction accounts for much of the importance of loopspaces in stable homotopy theory.

44.1 Relation between homotopy groups of a space and those of its loopspace

The basic relation between the homotopy groups is πk(X) ≊ πk−1(ΩX) .[1]

More generally,

[ΣZ,X] ≊ [Z,ΩX]

where, [A,B] is the set of homotopy classes of maps A → B , and ΣA is the suspension of A. In general [A,B]does not have a group structure for arbitrary spaces A and B . However, it can be shown that [ΣZ,X] and [Z,ΩX]do have natural group structures when Z and X are pointed, and the aforesaid isomorphism is of those groups. [2]

Note that setting Z = Sk−1 (the k − 1 sphere) gives the earlier result.

44.2 See also• fundamental group

• path (topology)

• loop group

• free loop

• quasigroup

99

100 CHAPTER 44. LOOP SPACE

44.3 References[1] http://topospaces.subwiki.org/wiki/Loop_space_of_a_based_topological_space

[2] May, J. P. (1999), “8”, A Concise Course in Algebraic Topology (PDF), U. Chicago Press, Chicago, retrieved 2008-09-27(chapter 8, section 2)

• Adams, John Frank (1978), Infinite loop spaces, Annals of Mathematics Studies 90, Princeton University Press,ISBN 978-0-691-08207-3, MR 505692

• May, J. Peter (1972), TheGeometry of Iterated Loop Spaces, Berlin, NewYork: Springer-Verlag, doi:10.1007/BFb0067491,ISBN 978-3-540-05904-2, MR 0420610

Chapter 45

Lower limit topology

In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set Rof real numbers; it is different from the standard topology on R (generated by the open intervals) and has a numberof interesting properties. It is the topology generated by the basis of all half-open intervals [a,b), where a and b arereal numbers.The resulting topological space, sometimes written Rl and called the Sorgenfrey line after Robert Sorgenfrey, oftenserves as a useful counterexample in general topology, like the Cantor set and the long line. The product of Rl withitself is also a useful counterexample, known as the Sorgenfrey plane.In complete analogy, one can also define the upper limit topology, or left half-open interval topology.

45.1 Properties• The lower limit topology is finer (has more open sets) than the standard topology on the real numbers (whichis generated by the open intervals). The reason is that every open interval can be written as a countably infiniteunion of half-open intervals.

• For any real a and b, the interval [a, b) is clopen in Rl (i.e., both open and closed). Furthermore, for all reala, the sets x ∈ R : x < a and x ∈ R : x ≥ a are also clopen. This shows that the Sorgenfrey line is totallydisconnected.

• Any compact subset of Rl must be a countable set. To see this, consider a non-empty compact subset C of Rl.Fix an x ∈ C, consider the following open cover of C:

[x,+∞)

∪(

−∞, x− 1n

) ∣∣∣n ∈ N.

Since C is compact, this cover has a finite subcover, and hence there exists a real number a(x) such thatthe interval (a(x), x] contains no point of C apart from x. This is true for all x ∈ C. Now choose a rationalnumber q(x) ∈ (a(x), x]. Since the intervals (a(x), x], parametrized by x ∈ C, are pairwise disjoint, thefunction q: C → Q is injective, and so C is a countable set.

• The name “lower limit topology” comes from the following fact: a sequence (or net) (xα) in Rl converges tothe limit L iff it “approaches L from the right”, meaning for every ε > 0 there exists an index α0 such that forall α > α0: L ≤ xα < L + ε. The Sorgenfrey line can thus be used to study right-sided limits: if f : R→ R is afunction, then the ordinary right-sided limit of f at x (when the codomain carry the standard topology) is thesame as the usual limit of f at x when the domain is equipped with the lower limit topology and the codomaincarries the standard topology.

• In terms of separation axioms, Rl is a perfectly normal Hausdorff space.

101

102 CHAPTER 45. LOWER LIMIT TOPOLOGY

• In terms of countability axioms, it is first-countable and separable, but not second-countable.

• In terms of compactness properties, Rl is Lindelöf and paracompact, but not σ-compact nor locally compact.

• Rl is not metrizable, since separable metric spaces are second-countable. However, the topology of a Sorgen-frey line is generated by a premetric.

• Rl is a Baire space .

45.2 References• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446

Chapter 46

Menger sponge

An illustration of M4, the sponge after four iterations of the construction process.

In mathematics, the Menger sponge (also known as the Menger universal curve) is a fractal curve. It is a three-dimensional generalization of the Cantor set and Sierpinski carpet. It was first described by Karl Menger in 1926, inhis studies of the concept of topological dimension.[1][2]

The Menger sponge simultaneously exhibits an infinite surface area and zero volume.[3]

46.1 Construction

The construction of a Menger sponge can be described as follows:

1. Begin with a cube (first image).

103

104 CHAPTER 46. MENGER SPONGE

A sculptural representation of iterations 0 (bottom) to 3 (top).

2. Divide every face of the cube into 9 squares, like a Rubik’s Cube. This will sub-divide the cube into 27 smallercubes.

3. Remove the smaller cube in the middle of each face, and remove the smaller cube in the very center of thelarger cube, leaving 20 smaller cubes (second image). This is a level-1 Menger sponge (resembling a VoidCube).

4. Repeat steps 2 and 3 for each of the remaining smaller cubes, and continue to iterate ad infinitum.

46.2. PROPERTIES 105

The second iteration will give you a level-2 sponge (third image), the third iteration gives a level-3 sponge (fourthimage), and so on. The Menger sponge itself is the limit of this process after an infinite number of iterations.

An illustration of the iterative construction of a Menger sponge up toM3, the third iteration.

The following table lists the properties of spongesMn where n is the number of iterations performed on the first cubeof unit side length.[4][5]

46.2 Properties

annotationsThe cross-section of a Menger sponge through its centroid and perpendicular to a space diagonal has sixfold symme-try containing hexagrams.[7]

Each face of the Menger sponge is a Sierpinski carpet; furthermore, any intersection of the Menger sponge with adiagonal or medium of the initial cube M0 is a Cantor set.The Menger sponge is a closed set; since it is also bounded, the Heine–Borel theorem implies that it is compact. Ithas Lebesgue measure 0. It is an uncountable set.The Lebesgue covering dimension of the Menger sponge is one, the same as any curve. Menger showed, in the 1926construction, that the sponge is a universal curve, in that every curve is homeomorphic to a subset of the Mengersponge, where a curve means any compact metric space of Lebesgue covering dimension one; this includes treesand graphs with an arbitrary countable number of edges, vertices and closed loops, connected in arbitrary ways. In asimilar way, the Sierpinski carpet is a universal curve for all curves that can be drawn on the two-dimensional plane.The Menger sponge constructed in three dimensions extends this idea to graphs that are not planar, and might beembedded in any number of dimensions.

106 CHAPTER 46. MENGER SPONGE

The Menger sponge has infinite surface area but zero volume.[3]

The sponge has a Hausdorff dimension of log 20/log 3 (approximately 2.726833).

46.3 Formal definition

Formally, a Menger sponge can be defined as follows:

M :=∩n∈N

Mn

where M0 is the unit cube and

Mn+1 :=

(x, y, z) ∈ R3 :

∃i, j, k ∈ 0, 1, 2 : (3x− i, 3y − j, 3z − k) ∈ Mn

and at most one of i, j, k is equal to 1

.

46.4 MegaMenger

A model of a tetrix viewed through the centre of the Cambridge Level-3 MegaMenger at the 2015 Cambridge Science Festival

MegaMenger is a project aiming to build the largest fractal model, pioneered byMatt Parker ofQueenMaryUniversityof London and Laura Taalman of James Madison University. Each small cube is made from 6 interlocking foldedbusiness cards, giving a total of 960 000 for a level-four sponge. The outer surfaces are then covered with paper orcardboard panels printed with a Sierpinski carpet design to be more aesthetically pleasing.[8] In 2014, twenty level-three Menger sponges were constructed, which combined would form a distributed level-four Menger sponge.[9]

46.5 Similar fractals• A Jerusalem Cube is a cube-based fractal with a Greek cross recursively removed from each face.[10]

46.6. SEE ALSO 107

One of the MegaMengers, at the University of Bath

• A Mosely snowflake is a cube-based fractal with corners recursively removed.[11]

• A tetrix is a tetrahedron-based fractal with tetrahedrons in its middle recursively removed.[12]

46.6 See also• Apollonian gasket

• Cantor cube

• Koch snowflake

• List of fractals by Hausdorff dimension

• Sierpiński tetrahedron

• Sierpiński triangle

46.7 Notes and references[1] Menger, Karl (1928), Dimensionstheorie, B.G Teubner Publishers

[2] Menger, Karl (1926), “Allgemeine Räume und Cartesische Räume. I.”, Communications to the Amsterdam Academy ofSciences. English translation reprinted in Edgar, Gerald A., ed. (2004), Classics on fractals, Studies in Nonlinearity,Westview Press. Advanced Book Program, Boulder, CO, ISBN 978-0-8133-4153-8, MR 2049443

[3] “Menger sponge”,Wolfram Alpha, retrieved 2013-12-12

[4] Wolfram Demonstrations Project, Volume and Surface Area of the Menger Sponge

[5] University of British Columbia Science and Mathematics Education Research Group, Mathematics Geometry: MengerSponge

108 CHAPTER 46. MENGER SPONGE

[6] http://nytimes.com/2011/06/28/science/28math-menger.html

[7] http://nytimes.com/2011/06/28/science/28math-menger.html

[8] Tim Chartier. “A Million Business Cards Present a Math Challenge”. Retrieved 2015-04-07.

[9] “MegaMenger”. Retrieved 2015-02-15.

[10] http://www.robertdickau.com/jerusalemcube.html

[11] http://wired.com/2012/09/folded-fractal-art-cards

[12] http://mathworld.wolfram.com/Tetrix.html

• Iwaniec, Tadeusz; Martin, Gaven (2001), Geometric function theory and non-linear analysis, Oxford Mathe-matical Monographs, The Clarendon Press Oxford University Press, ISBN 978-0-19-850929-5, MR 1859913.

• Zhou, Li (2007), “Problem 11208: Chromatic numbers of the Menger sponges”, American MathematicalMonthly 114 (9): 842

46.8 External links• Menger sponge at Wolfram MathWorld

• The 'Business CardMenger Sponge' by Dr. JeannineMosely – an online exhibit about this giant origami fractalat the Institute For Figuring

• An interactive Menger sponge

• Interactive Java models

• Puzzle Hunt — Video explaining Zeno’s paradoxes using Menger–Sierpinski sponge

• Menger Sponge Animations — Menger sponge animations up to level 9, discussion of optimization for 3d.

• Menger sphere, rendered in SunFlow

• Post-It Menger Sponge – a level-3 Menger sponge being built from Post-its

• The Mystery of the Menger Sponge. Sliced diagonally to reveal stars

• Number of cards required to build a Menger sponge of level n in origami

• Woolly Thoughts Level 2 Menger Sponge by two “Mathekniticians”

Chapter 47

Metric space

In mathematics, a metric space is a set for which distances between all members of the set are defined. Thosedistances, taken together, are called a metric on the set.The most familiar metric space is 3-dimensional Euclidean space. In fact, a “metric” is the generalization of theEuclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metricdefines the distance between two points as the length of the straight line segment connecting them. Other metricspaces occur for example in elliptic geometry and hyperbolic geometry, where distance on a sphere measured byangle is a metric, and the hyperboloid model of hyperbolic geometry is used by special relativity as a metric space ofvelocities.A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstracttopological spaces.

47.1 History

Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel, Rendic. Circ. Mat.Palermo 22 (1906) 1–74.

47.2 Definition

Ametric space is an ordered pair (M,d) whereM is a set and d is a metric onM , i.e., a function

d : M ×M → R

such that for any x, y, z ∈ M , the following holds:[1]

1. d(x, y) ≥ 0 (non-negative),

2. d(x, y) = 0 ⇐⇒ x = y (identity of indiscernibles),

3. d(x, y) = d(y, x) (symmetry) and

4. d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality) .

The first condition follows from the other three, since: for any x, y ∈ M ,

d(x, y) + d(y, x) ≥ d(x, x) inequality) triangle (by=⇒ d(x, y) + d(x, y) ≥ d(x, x) symmetry) (by=⇒ 2d(x, y) ≥ 0 indiscernibles) of identity (by=⇒ d(x, y) ≥ 0.

109

110 CHAPTER 47. METRIC SPACE

The function d is also called distance function or simply distance. Often, d is omitted and one just writes M for ametric space if it is clear from the context what metric is used.Ignoring mathematical details, for any system of roads and terrains the distance between two locations can be definedas the length of the shortest route connecting those locations. To be a metric there shouldn't be any one-way roads.The triangle inequality expresses the fact that detours aren't shortcuts. Many of the examples below can be seen asconcrete versions of this general idea.

47.3 Examples of metric spaces

• The real numbers with the distance function d(x, y) = |y − x| given by the absolute difference, and moregenerally Euclidean n -space with the Euclidean distance, are complete metric spaces. The rational numberswith the same distance also form a metric space, but are not complete.

• The positive real numbers with distance function d(x, y) = | log(y/x)| is a complete metric space.

• Any normed vector space is a metric space by defining d(x, y) = ∥y − x∥ , see also metrics on vector spaces.(If such a space is complete, we call it a Banach space.) Examples:

• The Manhattan norm gives rise to the Manhattan distance, where the distance between any two points,or vectors, is the sum of the differences between corresponding coordinates.

• The maximum norm gives rise to the Chebyshev distance or chessboard distance, the minimal numberof moves a chess king would take to travel from x to y .

• The British Rail metric (also called the Post Office metric or the SNCF metric) on a normed vector space isgiven by d(x, y) = ∥x∥ + ∥y∥ for distinct points x and y , and d(x, x) = 0 . More generally ∥.∥ can bereplaced with a function f taking an arbitrary set S to non-negative reals and taking the value 0 at most once:then the metric is defined on S by d(x, y) = f(x) + f(y) for distinct points x and y , and d(x, x) = 0 . Thename alludes to the tendency of railway journeys (or letters) to proceed via London (or Paris) irrespective oftheir final destination.

• If (M,d) is a metric space and X is a subset of M , then (X, d) becomes a metric space by restricting thedomain of d to X ×X .

• The discrete metric, where d(x, y) = 0 if x = y and d(x, y) = 1 otherwise, is a simple but importantexample, and can be applied to all sets. This, in particular, shows that for any set, there is always a metric spaceassociated to it. Using this metric, any point is an open ball, and therefore every subset is open and the spacehas the discrete topology.

• A finite metric space is a metric space having a finite number of points. Not every finite metric space can beisometrically embedded in a Euclidean space.[2][3]

• The hyperbolic plane is a metric space. More generally:

• IfM is any connected Riemannian manifold, then we can turnM into a metric space by defining the distanceof two points as the infimum of the lengths of the paths (continuously differentiable curves) connecting them.

• If X is some set and M is a metric space, then, the set of all bounded functions f : X → M (i.e. thosefunctions whose image is a bounded subset of M ) can be turned into a metric space by defining d(f, g) =supx∈X d(f(x), g(x)) for any two bounded functions f and g (where sup is supremum).[4] This metric is calledthe uniform metric or supremum metric, and If M is complete, then this function space is complete as well.If X is also a topological space, then the set of all bounded continuous functions fromX toM (endowed withthe uniform metric), will also be a complete metric if M is.

• If G is an undirected connected graph, then the set V of vertices of G can be turned into a metric space bydefining d(x, y) to be the length of the shortest path connecting the vertices x and y . In geometric grouptheory this is applied to the Cayley graph of a group, yielding the word metric.

• The Levenshtein distance is a measure of the dissimilarity between two strings u and v , defined as the minimalnumber of character deletions, insertions, or substitutions required to transform u into v . This can be thoughtof as a special case of the shortest path metric in a graph and is one example of an edit distance.

47.4. OPEN AND CLOSED SETS, TOPOLOGY AND CONVERGENCE 111

• Given a metric space (X, d) and an increasing concave function f : [0,∞) → [0,∞) such that f(x) = 0 ifand only if x = 0 , then f d is also a metric on X .

• Given an injective function f from any set A to a metric space (X, d) , d(f(x), f(y)) defines a metric on A .

• Using T-theory, the tight span of a metric space is also a metric space. The tight span is useful in several typesof analysis.

• The set of allm by n matrices over some field is a metric space with respect to the rank distance d(X,Y ) =rank(Y −X) .

• The Helly metric is used in game theory.

47.4 Open and closed sets, topology and convergence

Every metric space is a topological space in a natural manner, and therefore all definitions and theorems about generaltopological spaces also apply to all metric spaces.About any point x in a metric spaceM we define the open ball of radius r > 0 (where r is a real number) aboutx as the set

B(x; r) = y ∈ M : d(x, y) < r.

These open balls form the base for a topology on M, making it a topological space.Explicitly, a subset U ofM is called open if for every x in U there exists an r > 0 such that B(x; r) is contained inU . The complement of an open set is called closed. A neighborhood of the point x is any subset ofM that containsan open ball about x as a subset.A topological space which can arise in this way from a metric space is called a metrizable space; see the article onmetrization theorems for further details.A sequence ( xn ) in a metric spaceM is said to converge to the limit x ∈ M iff for every ϵ > 0 , there exists a naturalnumber N such that d(xn, x) < ϵ for all n > N . Equivalently, one can use the general definition of convergenceavailable in all topological spaces.A subset A of the metric spaceM is closed iff every sequence in A that converges to a limit inM has its limit in A .

47.5 Types of metric spaces

47.5.1 Complete spaces

Main article: Complete metric space

Ametric spaceM is said to be complete if every Cauchy sequence converges inM . That is to say: if d(xn, xm) → 0as both n andm independently go to infinity, then there is some y ∈ M with d(xn, y) → 0 .Every Euclidean space is complete, as is every closed subset of a complete space. The rational numbers, using theabsolute value metric d(x, y) = |x− y| , are not complete.Every metric space has a unique (up to isometry) completion, which is a complete space that contains the given spaceas a dense subset. For example, the real numbers are the completion of the rationals.IfX is a complete subset of the metric spaceM , thenX is closed inM . Indeed, a space is complete iff it is closedin any containing metric space.Every complete metric space is a Baire space.

112 CHAPTER 47. METRIC SPACE

A

diam(A)

Diameter of a set.

47.5.2 Bounded and totally bounded spaces

See also: bounded set

A metric space M is called bounded if there exists some number r, such that d(x,y) ≤ r for all x and y in M. Thesmallest possible such r is called the diameter of M. The space M is called precompact or totally bounded if forevery r > 0 there exist finitely many open balls of radius r whose union coversM. Since the set of the centres of theseballs is finite, it has finite diameter, from which it follows (using the triangle inequality) that every totally boundedspace is bounded. The converse does not hold, since any infinite set can be given the discrete metric (one of theexamples above) under which it is bounded and yet not totally bounded.Note that in the context of intervals in the space of real numbers and occasionally regions in a Euclidean space Rn

47.6. TYPES OF MAPS BETWEEN METRIC SPACES 113

a bounded set is referred to as “a finite interval” or “finite region”. However boundedness should not in general beconfused with “finite”, which refers to the number of elements, not to how far the set extends; finiteness impliesboundedness, but not conversely. Also note that an unbounded subset of Rn may have a finite volume.

47.5.3 Compact spaces

Ametric spaceM is compact if every sequence inM has a subsequence that converges to a point inM. This is knownas sequential compactness and, in metric spaces (but not in general topological spaces), is equivalent to the topologicalnotions of countable compactness and compactness defined via open covers.Examples of compact metric spaces include the closed interval [0,1] with the absolute value metric, all metric spaceswith finitely many points, and the Cantor set. Every closed subset of a compact space is itself compact.A metric space is compact iff it is complete and totally bounded. This is known as the Heine–Borel theorem. Notethat compactness depends only on the topology, while boundedness depends on the metric.Lebesgue’s number lemma states that for every open cover of a compact metric space M, there exists a “Lebesguenumber” δ such that every subset of M of diameter < δ is contained in some member of the cover.Every compact metric space is second countable,[5] and is a continuous image of the Cantor set. (The latter result isdue to Pavel Alexandrov and Urysohn.)

47.5.4 Locally compact and proper spaces

A metric space is said to be locally compact if every point has a compact neighborhood. Euclidean spaces are locallycompact, but infinite-dimensional Banach spaces are not.A space is proper if every closed ball y : d(x,y) ≤ r is compact. Proper spaces are locally compact, but the converseis not true in general.

47.5.5 Connectedness

A metric spaceM is connected if the only subsets that are both open and closed are the empty set andM itself.A metric space M is path connected if for any two points x, y ∈ M there exists a continuous map f : [0, 1] → Mwith f(0) = x and f(1) = y . Every path connected space is connected, but the converse is not true in general.There are also local versions of these definitions: locally connected spaces and locally path connected spaces.Simply connected spaces are those that, in a certain sense, do not have “holes”.

47.5.6 Separable spaces

A metric space is separable space if it has a countable dense subset. Typical examples are the real numbers orany Euclidean space. For metric spaces (but not for general topological spaces) separability is equivalent to secondcountability and also to the Lindelöf property.

47.6 Types of maps between metric spaces

Suppose (M1,d1) and (M2,d2) are two metric spaces.

47.6.1 Continuous maps

Main article: Continuous function (topology)

The map f:M1→M2 is continuous if it has one (and therefore all) of the following equivalent properties:

114 CHAPTER 47. METRIC SPACE

General topological continuity for every open set U in M2, the preimage f −1(U) is open in M1

This is the general definition of continuity in topology.

Sequential continuity if (xn) is a sequence in M1 that converges to x in M1, then the sequence (f(xn)) convergesto f(x) in M2.

This is sequential continuity, due to Eduard Heine.

ε-δ definition for every x in M1 and every ε>0 there exists δ>0 such that for all y in M1 we have

d1(x, y) < δ ⇒ d2(f(x), f(y)) < ε.

This uses the (ε, δ)-definition of limit, and is due to Augustin Louis Cauchy.

Moreover, f is continuous if and only if it is continuous on every compact subset of M1.The image of every compact set under a continuous function is compact, and the image of every connected set undera continuous function is connected.

47.6.2 Uniformly continuous maps

The map ƒ : M1 → M2 is uniformly continuous if for every ε > 0 there exists δ > 0 such that

d1(x, y) < δ ⇒ d2(f(x), f(y)) < ε for all x, y ∈ M1.

Every uniformly continuous map ƒ : M1 →M2 is continuous. The converse is true ifM1 is compact (Heine–Cantortheorem).Uniformly continuous maps turn Cauchy sequences in M1 into Cauchy sequences in M2. For continuous maps thisis generally wrong; for example, a continuous map from the open interval (0,1) onto the real line turns some Cauchysequences into unbounded sequences.

47.6.3 Lipschitz-continuous maps and contractions

Given a number K > 0, the map ƒ : M1 → M2 is K-Lipschitz continuous if

d2(f(x), f(y)) ≤ Kd1(x, y) for all x, y ∈ M1.

Every Lipschitz-continuous map is uniformly continuous, but the converse is not true in general.If K < 1, then ƒ is called a contraction. SupposeM2 =M1 andM1 is complete. If ƒ is a contraction, then ƒ admits aunique fixed point (Banach fixed point theorem). If M1 is compact, the condition can be weakened a bit: ƒ admits aunique fixed point if

d(f(x), f(y)) < d(x, y) for all x = y ∈ M1

47.6.4 Isometries

The map f:M1→M2 is an isometry if

d2(f(x), f(y)) = d1(x, y) for all x, y ∈ M1

Isometries are always injective; the image of a compact or complete set under an isometry is compact or complete,respectively. However, if the isometry is not surjective, then the image of a closed (or open) set need not be closed(or open).

47.7. NOTIONS OF METRIC SPACE EQUIVALENCE 115

47.6.5 Quasi-isometries

The map f : M1 → M2 is a quasi-isometry if there exist constants A ≥ 1 and B ≥ 0 such that

1

Ad2(f(x), f(y))−B ≤ d1(x, y) ≤ Ad2(f(x), f(y)) +B all for x, y ∈ M1

and a constant C ≥ 0 such that every point in M2 has a distance at most C from some point in the image f(M1).Note that a quasi-isometry is not required to be continuous. Quasi-isometries compare the “large-scale structure” ofmetric spaces; they find use in geometric group theory in relation to the word metric.

47.7 Notions of metric space equivalence

Given two metric spaces (M1, d1) and (M2, d2):

• They are called homeomorphic (topologically isomorphic) if there exists a homeomorphism between them(i.e., a bijection continuous in both directions).

• They are called uniformic (uniformly isomorphic) if there exists a uniform isomorphism between them (i.e.,a bijection uniformly continuous in both directions).

• They are called isometric if there exists a bijective isometry between them. In this case, the two metric spacesare essentially identical.

• They are called quasi-isometric if there exists a quasi-isometry between them.

47.8 Topological properties

Metric spaces are paracompact[6] Hausdorff spaces[7] and hence normal (indeed they are perfectly normal). Animportant consequence is that every metric space admits partitions of unity and that every continuous real-valuedfunction defined on a closed subset of a metric space can be extended to a continuous map on the whole space (Tietzeextension theorem). It is also true that every real-valued Lipschitz-continuous map defined on a subset of a metricspace can be extended to a Lipschitz-continuous map on the whole space.Metric spaces are first countable since one can use balls with rational radius as a neighborhood base.The metric topology on a metric spaceM is the coarsest topology onM relative to which the metric d is a continuousmap from the product of M with itself to the non-negative real numbers.

47.9 Distance between points and sets; Hausdorff distance and Gromovmetric

A simple way to construct a function separating a point from a closed set (as required for a completely regular space)is to consider the distance between the point and the set. If (M,d) is a metric space, S is a subset ofM and x is a pointof M, we define the distance from x to S as

d(x, S) = infd(x, s) : s ∈ S where inf represents the infimum.

Then d(x, S) = 0 if and only if x belongs to the closure of S. Furthermore, we have the following generalization of thetriangle inequality:

d(x, S) ≤ d(x, y) + d(y, S),

116 CHAPTER 47. METRIC SPACE

which in particular shows that the map x 7→ d(x, S) is continuous.Given two subsets S and T of M, we define their Hausdorff distance to be

dH(S, T ) = maxsupd(s, T ) : s ∈ S, supd(t, S) : t ∈ T where sup represents the supremum.

In general, the Hausdorff distance dH(S,T) can be infinite. Two sets are close to each other in the Hausdorff distanceif every element of either set is close to some element of the other set.The Hausdorff distance dH turns the set K(M) of all non-empty compact subsets of M into a metric space. One canshow that K(M) is complete ifM is complete. (A different notion of convergence of compact subsets is given by theKuratowski convergence.)One can then define the Gromov–Hausdorff distance between any two metric spaces by considering the minimalHausdorff distance of isometrically embedded versions of the two spaces. Using this distance, the class of all (isometryclasses of) compact metric spaces becomes a metric space in its own right.

47.10 Product metric spaces

If (M1, d1), . . . , (Mn, dn) aremetric spaces, andN is the Euclidean norm onRn, then(M1×. . .×Mn, N(d1, . . . , dn)

)is a metric space, where the product metric is defined by

N(d1, ..., dn)((x1, . . . , xn), (y1, . . . , yn)

)= N

(d1(x1, y1), . . . , dn(xn, yn)

),

and the induced topology agrees with the product topology. By the equivalence of norms in finite dimensions, anequivalent metric is obtained if N is the taxicab norm, a p-norm, the max norm, or any other norm which is non-decreasing as the coordinates of a positive n-tuple increase (yielding the triangle inequality).Similarly, a countable product of metric spaces can be obtained using the following metric

d(x, y) =

∞∑i=1

1

2idi(xi, yi)

1 + di(xi, yi).

An uncountable product of metric spaces need not be metrizable. For example, RR is not first-countable and thusisn't metrizable.

47.10.1 Continuity of distance

It is worth noting that in the case of a single space (M,d) , the distance map d : M ×M → R+ (from the definition)is uniformly continuous with respect to any of the above product metrics N(d, d) , and in particular is continuouswith respect to the product topology ofM ×M .

47.11 Quotient metric spaces

If M is a metric space with metric d, and ~ is an equivalence relation on M, then we can endow the quotient set M/~with the following (pseudo)metric. Given two equivalence classes [x] and [y], we define

d′([x], [y]) = infd(p1, q1) + d(p2, q2) + · · ·+ d(pn, qn)

where the infimum is taken over all finite sequences (p1, p2, . . . , pn) and (q1, q2, . . . , qn) with [p1] = [x] , [qn] = [y], [qi] = [pi+1], i = 1, 2, . . . , n − 1 . In general this will only define a pseudometric, i.e. d′([x], [y]) = 0 doesnot necessarily imply that [x] = [y] . However, for nice equivalence relations (e.g., those given by gluing togetherpolyhedra along faces), it is a metric.

47.12. GENERALIZATIONS OF METRIC SPACES 117

The quotient metric d is characterized by the following universal property. If f : (M,d) −→ (X, δ) is a metric mapbetween metric spaces (that is, δ(f(x), f(y)) ≤ d(x, y) for all x, y) satisfying f(x)=f(y) whenever x ∼ y, then theinduced function f : M/ ∼−→ X , given by f([x]) = f(x) , is a metric map f : (M/ ∼, d′) −→ (X, δ).

A topological space is sequential if and only if it is a quotient of a metric space.[8]

47.12 Generalizations of metric spaces

• Every metric space is a uniform space in a natural manner, and every uniform space is naturally a topologicalspace. Uniform and topological spaces can therefore be regarded as generalizations of metric spaces.

• If we consider the first definition of a metric space given above and relax the second requirement, we arriveat the concepts of a pseudometric space or a dislocated metric space.[9] If we remove the third or fourth, wearrive at a quasimetric space, or a semimetric space.

• If the distance function takes values in the extended real number line R∪+∞, but otherwise satisfies all fourconditions, then it is called an extended metric and the corresponding space is called an ∞ -metric space. If thedistance function takes values in some (suitable) ordered set (and the triangle inequality is adjusted accordingly),then we arrive at the notion of generalized ultrametric.[10]

• Approach spaces are a generalization ofmetric spaces, based on point-to-set distances, instead of point-to-pointdistances.

• A continuity space is a generalization of metric spaces and posets, that can be used to unify the notions ofmetric spaces and domains.

• A partial metric space is intended to be the least generalisation of the notion of a metric space, such that thedistance of each point from itself is no longer necessarily zero.[11]

47.12.1 Metric spaces as enriched categories

The ordered set (R,≥) can be seen as a category by requesting exactly one morphism a → b if a ≥ b and noneotherwise. By using+ as the tensor product and 0 as the identity, it becomes a monoidal category R∗ . Every metricspace (M,d) can now be viewed as a categoryM∗ enriched over R∗ :

• Set Ob(M∗) := M

• For each X,Y ∈ M set Hom(X,Y ) := d(X,Y ) ∈ Ob(R∗)

• The composition morphism Hom(Y, Z) ⊗ Hom(X,Y ) → Hom(X,Z) will be the unique morphism in R∗

given from the triangle inequality d(y, z) + d(x, y) ≥ d(x, z)

• The identity morphism 0 → Hom(X,X) will be the unique morphism given from the fact that 0 ≥ d(X,X) .

• Since R∗ is a poset, all diagrams that are required for an enriched category commute automatically.

See the paper by F.W. Lawvere listed below.

47.13 See also

• Space (mathematics)

• Metric (mathematics)

• Metric signature

118 CHAPTER 47. METRIC SPACE

• Metric tensor

• Metric tree

• Norm (mathematics)

• Normed vector space

• Measure (mathematics)

• Hilbert space

• Product metric

• Aleksandrov–Rassias problem

• Category of metric spaces

• Classical Wiener space

• Glossary of Riemannian and metric geometry

• Isometry, contraction mapping and metric map

• Lipschitz continuity

• Triangle inequality

47.14 Notes

[1] B. Choudhary (1992). The Elements of Complex Analysis. New Age International. p. 20. ISBN 978-81-224-0399-2.

[2] Nathan Linial. Finite Metric Spaces—Combinatorics, Geometry and Algorithms, Proceedings of the ICM, Beijing 2002,vol. 3, pp573–586

[3] Open problems on embeddings of finite metric spaces, edited by Jirīı Matoušek, 2007

[4] Searcóid, p. 107.

[5] PlanetMath: a compact metric space is second countable

[6] Rudin, Mary Ellen. A new proof that metric spaces are paracompact. Proceedings of the American Mathematical Society,Vol. 20, No. 2. (Feb., 1969), p. 603.

[7] metric spaces are Hausdorff at PlanetMath.org.

[8] Goreham, Anthony. Sequential convergence in Topological Spaces. Honours’ Dissertation, Queen’s College, Oxford (April,2001), p. 14

[9] Pascal Hitzler and Anthony Seda, Mathematical Aspects of Logic Programming Semantics. Chapman and Hall/CRC,2010.

[10] Pascal Hitzler and Anthony Seda, Mathematical Aspects of Logic Programming Semantics. Chapman and Hall/CRC,2010.

[11] http://www.dcs.warwick.ac.uk/pmetric/

47.15. REFERENCES 119

47.15 References• Victor Bryant, Metric Spaces: Iteration and Application, Cambridge University Press, 1985, ISBN 0-521-31897-1.

• Dmitri Burago, Yu D Burago, Sergei Ivanov, A Course in Metric Geometry, American Mathematical Society,2001, ISBN 0-8218-2129-6.

• Athanase Papadopoulos,Metric Spaces, Convexity and Nonpositive Curvature, European Mathematical Society,First edition 2004, ISBN 978-3-03719-010-4. Second edition 2014, ISBN 978-3-03719-132-3.

• Mícheál Ó Searcóid,Metric Spaces, Springer Undergraduate Mathematics Series, 2006, ISBN 1-84628-369-8.

• Lawvere, F. William, “Metric spaces, generalized logic, and closed categories”, [Rend. Sem. Mat. Fis. Milano43 (1973), 135—166 (1974); (Italian summary)

This is reprinted (with author commentary) at Reprints in Theory and Applications of Categories Also (with an authorcommentary) in Enriched categories in the logic of geometry and analysis. Repr. Theory Appl. Categ. No. 1 (2002),1–37.

• Weisstein, Eric W., “Product Metric”, MathWorld.

47.16 External links• Hazewinkel, Michiel, ed. (2001), “Metric space”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Far and near — several examples of distance functions at cut-the-knot.

Chapter 48

Moore plane

In mathematics, theMoore plane, also sometimes called Niemytzki plane (or Nemytskii plane, Nemytskii’s tan-gent disk topology), is a topological space. It is a completely regular Hausdorff space (also called Tychonoff space)which is not normal. It is named after Robert Lee Moore and Viktor Vladimirovich Nemytskii.

48.1 Definition

Open neighborhood of the Niemytzki plane, tangent to the x-axis

If Γ is the upper half-plane Γ = (x, y) ∈ R2|y ≥ 0 , then a topology may be defined on Γ by taking a local basis

120

48.2. PROPERTIES 121

B(p, q) as follows:

• Elements of the local basis at points (x, y)with y > 0 are the open discs in the plane which are small enough tolie within Γ . Thus the subspace topology inherited by Γ\(x, 0)|x ∈ R is the same as the subspace topologyinherited from the standard topology of the Euclidean plane.

• Elements of the local basis at points p = (x, 0) are sets p∪Awhere A is an open disc in the upper half-planewhich is tangent to the x axis at p.

That is, the local basis is given by

B(p, q) =

Uϵ(p, q) := (x, y) : (x− p)2 + (y − q)2 < ϵ2 | ϵ > 0, if q > 0;

Vϵ(p) := (p, 0) ∪ (x, y) : (x− p)2 + (y − ϵ)2 < ϵ2 | ϵ > 0, if q = 0.

48.2 Properties

• The Moore plane Γ is separable, that is, it has a countable dense subset.

• The Moore plane is a completely regular Hausdorff space (i.e. Tychonoff space), which is not normal.

• The subspace (x, 0) ∈ Γ|x ∈ R of Γ has, as its subspace topology, the discrete topology. Thus, the Mooreplane shows that a subspace of a separable space need not be separable.

• The Moore plane is first countable, but not second countable or Lindelöf.

• The Moore plane is not locally compact.

• The Moore plane is countably metacompact but not metacompact.

48.3 Proof that the Moore plane is not normal

The fact that this spaceM is not normal can be established by the following counting argument (which is very similarto the argument that the Sorgenfrey plane is not normal):

1. On the one hand, the countable set S := (p, q) ∈ Q × Q : q > 0 of points with rational coordinates isdense in M; hence every continuous function f : M → R is determined by its restriction to S , so there canbe at most |R||S| = 2ℵ0 many continuous real-valued functions on M.

2. On the other hand, the real lineL := (p, 0) : p ∈ R is a closed discrete subspace ofM with 2ℵ0 many points.So there are 22ℵ0

> 2ℵ0 many continuous functions from L to R . Not all these functions can be extended tocontinuous functions on M.

3. Hence M is not normal, because by the Tietze extension theorem all continuous functions defined on a closedsubspace of a normal space can be extended to a continuous function on the whole space.

In fact, if X is a separable topological space having an uncountable closed discrete subspace, X cannot be normal.

48.4 See also

• Moore space

• Hedgehog space

122 CHAPTER 48. MOORE PLANE

48.5 References• Stephen Willard. General Topology, (1970) Addison-Wesley ISBN 0-201-08707-3.

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446 (Example 82)

• Niemytzki plane at PlanetMath.org.

Chapter 49

Nilpotent space

In topology, a branch of mathematics, a nilpotent space is a based topological space X such that the fundamentalgroup π = π1X is a nilpotent group and π acts nilpotently on higher homotopy groups πiX, i ≥ 2 .[1] A simplyconnected space and a simple space are (trivial) examples of nilpotent spaces.

49.1 References[1] Bousfield, A. K.; Kan, D. M. (1987), Homotopy Limits, Completions and Localizations, Lecture Notes in Mathematics 304,

Springer, p. 59, ISBN 9783540061052.

123

Chapter 50

Overlapping interval topology

Not to be confused with Interlocking interval topology.

In mathematics, the overlapping interval topology is a topology which is used to illustrate various topological prin-ciples.

50.1 Definition

Given the closed interval [−1, 1] of the real number line, the open sets of the topology are generated from the half-open intervals [−1, b) and (a, 1] with a < 0 < b . The topology therefore consists of intervals of the form [−1, b) ,(a, b) , and (a, 1] with a < 0 < b , together with [−1, 1] itself and the empty set.

50.2 Properties

Any two distinct points in [−1, 1] are topologically distinguishable under the overlapping interval topology as one canalways find an open set containing one but not the other point. However, every non-empty open set contains the point0 which can therefore not be separated from any other point in [−1, 1] , making [−1, 1] with the overlapping intervaltopology an example of a T0 space that is not a T1 space.The overlapping interval topology is second countable, with a countable basis being given by the intervals [−1, s) ,(r, s) and (r, 1] with r < 0 < s and r and s rational (and thus countable).

50.3 References• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446 (See example 53)

124

Chapter 51

Partially ordered space

In mathematics, a partially ordered space (or pospace) is a topological spaceX equipped with a closed partial order≤ , i.e. a partial order whose graph (x, y) ∈ X2|x ≤ y is a closed subset of X2 .From pospaces, one can define dimaps, i.e. continuous maps between pospaces which preserve the order relation.

51.1 Equivalences

For a topological spaceX equipped with a partial order ≤ , the following are equivalent:

• X is a partially ordered space.

• For all x, y ∈ X with x ≤ y , there are open setsU, V ⊂ X with x ∈ U, y ∈ V and u ≤ v for all u ∈ U, v ∈ V.

• For all x, y ∈ X with x ≤ y , there are disjoint neighbourhoods U of x and V of y such that U is an upper setand V is a lower set.

The order topology is a special case of this definition, since a total order is also a partial order. Every pospace isa Hausdorff space. If we take equality = as the partial order, this definition becomes the definition of a Hausdorffspace.

51.2 See also• Ordered vector space

51.3 External link• ordered space on Planetmath

125

Chapter 52

Particular point topology

In mathematics, the particular point topology (or included point topology) is a topology where sets are consideredopen if they are empty or contain a particular, arbitrarily chosen, point of the topological space. Formally, let X beany set and p ∈ X. The collection

T = S ⊆ X: p ∈ S or S = ∅

of subsets of X is then the particular point topology on X. There are a variety of cases which are individually named:

• If X = 0,1 we call X the Sierpiński space. This case is somewhat special and is handled separately.• If X is finite (with at least 3 points) we call the topology on X the finite particular point topology.• If X is countably infinite we call the topology on X the countable particular point topology.• If X is uncountable we call the topology on X the uncountable particular point topology.

A generalization of the particular point topology is the closed extension topology. In the case when X \ p has thediscrete topology, the closed extension topology is the same as the particular point topology.This topology is used to provide interesting examples and counterexamples.

52.1 PropertiesClosed sets have empty interior Given an open set A ⊂ X every x = p is a limit point of A. So the closure of

any open set other than ∅ isX . No closed set other thanX contains p so the interior of every closed set otherthan X is ∅ .

52.1.1 Connectedness PropertiesPath and locally connected but not arc connected

f(t) =

x t = 0

p t ∈ (0, 1)

y t = 1

f is a path for all x,y ∈ X. However since p is open, the preimage of p under a continuous injection from[0,1] would be an open single point of [0,1], which is a contradiction.

Dispersion point, example of a set with p is a dispersion point for X. That is X\p is totally disconnected.

Hyperconnected but not ultraconnected Every open set contains p hence X is hyperconnected. But if a and b arein X such that p, a, and b are three distinct points, then a and b are disjoint closed sets and thus X is notultraconnected. Note that if X is the Sierpinski space then no such a and b exist and X is in fact ultraconnected.

126

52.1. PROPERTIES 127

52.1.2 Compactness Properties

Closure of compact not compact The set p is compact. However its closure (the closure of a compact set) is theentire space X and if X is infinite this is not compact (since any set t,p is open). For similar reasons if X isuncountable then we have an example where the closure of a compact set is not a Lindelöf space.

Pseudocompact but not weakly countably compact First there are no disjoint non-empty open sets (since all opensets contain 'p'). Hence every continuous function to the real line must be constant, and hence bounded, provingthat X is a pseudocompact space. Any set not containing p does not have a limit point thus if X if infinite it isnot weakly countably compact.

Locally compact but not strongly locally compact. Both possibilities regarding global compactness.If x ∈ X then the set x, p is a compact neighborhood of x. However the closure of this neighborhood is allof X and hence X is not strongly locally compact.

In terms of global compactness, X finite if and only if X is compact. The first implication is immediate, the reverseimplication follows from noting that

∪x∈Xp, x is an open cover with no finite subcover.

52.1.3 Limit related

Accumulation point but not a ω-accumulation point If Y is some subset containing p then any x different fromp is an accumulation point of Y. However x is not an ω-accumulation point as x,p is one neighbourhood of xwhich does not contain infinitely many points from Y. Because this makes no use of properties of Y it leads tooften cited counter examples.

Accumulation point as a set but not as a sequence Take a sequence ai of distinct elements that also contains p.As in the example above, the underlying set has any x different from p as an accumulation point. However thesequence itself cannot possess accumulation point y for its neighbourhood y,p must contain infinite numberof the distinct ai.

52.1.4 Separation related

T0 X is T0 (since x, p is open for each x) but satisfies no higher separation axioms (because all open sets mustcontain p).

Not regular Since every nonempty open set contains p, no closed set not containing p (such as X\p) can beseparated by neighbourhoods from p, and thus X is not regular. Since complete regularity implies regu-larity, X is not completely regular.

Not normal Since every nonempty open set contains p, no nonempty closed sets can be separated by neighbourhoodsfrom each other, and thus X is not normal. Exception: the Sierpinski topology is normal, and even completelynormal, since it contains no nontrivial separated sets.

Separability p is dense and henceX is a separable space. However ifX is uncountable thenX\p is not separable.This is an example of a subspace of a separable space not being separable.

Countability (first but not second) If X is uncountable then X is first countable but not second countable.

Comparable ( Homeomorphic topology on the same set that is not comparable)Let p, q ∈ X with p = q . Let tp = S ⊂ X | p ∈ S and tq = S ⊂ X | q ∈ S . That is tq is the par-ticular point topology on X with q being the distinguished point. Then (X,tp) and (X,tq) are homeomorphicincomparable topologies on the same set.

Density (no nonempty subsets dense in themselves) Let S be a subset of X. If S contains p then S has no limitpoints (see limit point section). If S does not contain p then p is not a limit point of S. Hence S is not dense ifS is nonempty.

128 CHAPTER 52. PARTICULAR POINT TOPOLOGY

Not first category Any set containing p is dense in X. Hence X is not a union of nowhere dense subsets.

Subspaces Every subspace of a set given the particular point topology that doesn't contain the particular point,inherits the discrete topology.

52.2 See also• Sierpiński space

• Excluded point topology

• Alexandrov topology

• Finite topological space

• One-point compactification

52.3 References• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446

Chapter 53

Partition topology

In mathematics, the partition topology is a topology that can be induced on any set X by partitioning X into disjointsubsets P; these subsets form the basis for the topology. There are two important examples which have their ownnames:

• The odd–even topology is the topology where X = N and P = 2k − 1, 2k, k ∈ N.

• The deleted integer topology is defined by lettingX =∪

n∈N(n− 1, n) ⊂ R andP = (0, 1), (1, 2), (2, 3), . . . .

The trivial partitions yield the discrete topology (each point of X is a set in P) or indiscrete topology ( P = X ).Any set X with a partition topology generated by a partition P can be viewed as a pseudometric space with a pseudo-metric given by:

d(x, y) =

0 ifx and ypartition same the in are1 otherwise,

This is not a metric unless P yields the discrete topology.The partition topology provides an important example of the independence of various separation axioms. Unless Pis trivial, at least one set in P contains more than one point, and the elements of this set are topologically indistin-guishable: the topology does not separate points. Hence X is not a Kolmogorov space, nor a T1 space, a Hausdorffspace or an Urysohn space. In a partition topology the complement of every open set is also open, and therefore aset is open if and only if it is closed. Therefore, X is a regular, completely regular, normal and completely normal.We note also that X/P is the discrete topology.

53.1 References• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446

129

Chapter 54

Pointed space

In mathematics, a pointed space is a topological space X with a distinguished basepoint x0 in X. Maps of pointedspaces (based maps) are continuous maps preserving basepoints, i.e. a continuous map f : X → Y such that f(x0) =y0. This is usually denoted

f : (X, x0) → (Y, y0).

Pointed spaces are important in algebraic topology, particularly in homotopy theory, where many constructions, suchas the fundamental group, depend on a choice of basepoint.The pointed set concept is less important; it is anyway the case of a pointed discrete space.

54.1 Category of pointed spaces

The class of all pointed spaces forms a category Top• with basepoint preserving continuous maps as morphisms.Another way to think about this category is as the comma category, (• ↓ Top) where • is any one point spaceand Top is the category of topological spaces. (This is also called a coslice category denoted •/Top.) Objectsin this category are continuous maps • → X. Such morphisms can be thought of as picking out a basepoint in X.Morphisms in (• ↓ Top) are morphisms in Top for which the following diagram commutes:

It is easy to see that commutativity of the diagram is equivalent to the condition that f preserves basepoints.As a pointed space • is a zero object in Top• while it is only a terminal object in Top.There is a forgetful functor Top• → Top which “forgets” which point is the basepoint. This functor has a left adjointwhich assigns to each topological space X the disjoint union of X and a one point space • whose single element istaken to be the basepoint.

54.2 Operations on pointed spaces• A subspace of a pointed space X is a topological subspace A ⊆ X which shares its basepoint with X so that theinclusion map is basepoint preserving.

• One can form the quotient of a pointed space X under any equivalence relation. The basepoint of the quotientis the image of the basepoint in X under the quotient map.

130

54.3. REFERENCES 131

• One can form the product of two pointed spaces (X, x0), (Y, y0) as the topological product X × Y with (x0,y0) serving as the basepoint.

• The coproduct in the category of pointed spaces is the wedge sum, which can be thought of as the one-pointunion of spaces.

• The smash product of two pointed spaces is essentially the quotient of the direct product and the wedge sum.The smash product turns the category of pointed spaces into a symmetric monoidal category with the pointed0-sphere as the unit object.

• The reduced suspension ΣX of a pointed space X is (up to a homeomorphism) the smash product of X andthe pointed circle S1.

• The reduced suspension is a functor from the category of pointed spaces to itself. This functor is a left adjointto the functor Ω taking a based space X to its loop space ΩX .

54.3 References• Gamelin, Theodore W.; Greene, Robert Everist (1999) [1983]. Introduction to Topology (second ed.). DoverPublications. ISBN 0-486-40680-6.

• Mac Lane, Saunders (September 1998). Categories for the Working Mathematician (second ed.). Springer.ISBN 0-387-98403-8.

• mathoverflow discussion on several base points and groupoids

Chapter 55

Pointwise convergence

In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to aparticular function.[1][2]

55.1 Definition

Suppose fn is a sequence of functions sharing the same domain and codomain (for the moment, defer specifyingthe nature of the values of these functions, but the reader may take them to be real numbers). The sequence fn converges pointwise to f, often written as

limn→∞

fn = f pointwise,

if and only if

limn→∞

fn(x) = f(x).

for every x in the domain.

55.2 Properties

This concept is often contrasted with uniform convergence. To say that

limn→∞

fn = f uniformly

means that

limn→∞

sup |fn(x)− f(x)| : x ∈ the domain = 0.

That is a stronger statement than the assertion of pointwise convergence: every uniformly convergent sequence ispointwise convergent, to the same limiting function, but some pointwise convergent sequences are not uniformlyconvergent. For example we have

limn→∞

xn = 0 pointwise on the interval [0, 1), but not uniformly on the interval [0, 1).

as the speed of convergence depends on x and is faster for lower values of x in the domain.

132

55.3. TOPOLOGY 133

The pointwise limit of a sequence of continuous functionsmay be a discontinuous function, but only if the convergenceis not uniform. For example,

f(x) = limn→∞

cos(πx)2n

takes the value 1 when x is an integer and 0 when x is not an integer, and so is discontinuous at every integer.The values of the functions fn need not be real numbers, but may be in any topological space, in order that the conceptof pointwise convergence make sense. Uniform convergence, on the other hand, does not make sense for functionstaking values in topological spaces generally, but makes sense for functions taking values in metric spaces, and, moregenerally, in uniform spaces.

55.3 Topology

Pointwise convergence is the same as convergence in the product topology on the space YX, where X is the domainand Y is the codomain. If the codomain Y is compact, then, by Tychonoff’s theorem, the space YX is also compact.

55.4 Almost everywhere convergence

In measure theory, one talks about almost everywhere convergence of a sequence of measurable functions defined ona measurable space. That means pointwise convergence almost everywhere. Egorov’s theorem states that pointwiseconvergence almost everywhere on a set of finite measure implies uniform convergence on a slightly smaller set.

55.5 See also• Modes of convergence (annotated index)

55.6 References[1] Rudin, Walter (1976). Principles of Mathematical Analysis. McGraw-Hill. ISBN 0-07-054235-X.

[2] Munkres, James R. (2000). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.

Chapter 56

Priestley space

In mathematics, a Priestley space is an ordered topological space with special properties. Priestley spaces are namedafter Hilary Priestley who introduced and investigated them.[1] Priestley spaces play a fundamental role in the studyof distributive lattices. In particular, there is a duality between the category of Priestley spaces and the category ofbounded distributive lattices.[2] [3]

56.1 Definition

A Priestley space is an ordered topological space (X,τ,≤), i. e. a set X equipped with a partial order ≤ and a topologyτ, satisfying the following two conditions: (i) (X,τ) is compact. (ii) If x ≤ y , then there exists a clopen up-set U of Xsuch that x∈U and y∉ U. (This condition is known as the Priestley separation axiom.)

56.2 Properties of Priestley spaces• Each Priestley space is Hausdorff. Indeed, given two points x,y of a Priestley space (X,τ,≤), if x≠ y, then as ≤is a partial order, either x ≤ y or y ≤ x . Assuming, without loss of generality, that x ≤ y , (ii) provides a clopenup-set U of X such that x∈ U and y∉ U. Therefore, U and V = X − U are disjoint open subsets of X separatingx and y.

• Each Priestley space is also zero-dimensional; that is, each open neighborhood U of a point x of a Priestleyspace (X,τ,≤) contains a clopen neighborhood C of x. To see this, one proceeds as follows. For each y ∈ X −U, either x ≤ y or y ≤ x . By the Priestley separation axiom, there exists a clopen up-set or a clopen down-setcontaining x and missing y. Obviously the intersection of these clopen neighborhoods of x does not meet X −U. Therefore, as X is compact, there exists a finite intersection of these clopen neighborhoods of x missing X− U. This finite intersection is the desired clopen neighborhood C of x contained in U.

It follows that for each Priestley space (X,τ,≤), the topological space (X,τ) is a Stone space; that is, it is a compactHausdorff zero-dimensional space.Some further useful properties of Priestley spaces are listed below.Let (X,τ,≤) be a Priestley space.

(a) For each closed subset F of X, both ↑ F = x ∈ X : y ≤ x for some y ∈ F and ↓ F = x ∈X : x ≤ y for some y ∈ F are closed subsets of X.

(b) Each open up-set of X is a union of clopen up-sets of X and each open down-set of X isa union of clopen down-sets of X.

(c) Each closed up-set ofX is an intersection of clopen up-sets ofX and each closed down-setof X is an intersection of clopen down-sets of X.

134

56.3. CONNECTION WITH SPECTRAL SPACES 135

(d) Clopen up-sets and clopen down-sets of X form a subbasis for (X,τ).

(e) For each pair of closed subsets F and G of X, if ↑F ∩ ↓G = ∅, then there exists a clopenup-set U such that F ⊆ U and U ∩ G = ∅.

A Priestley morphism from a Priestley space (X,τ,≤) to another Priestley space (X′,τ′,≤′) is a map f : X → X′ whichis continuous and order-preserving.Let Pries denote the category of Priestley spaces and Priestley morphisms.

56.3 Connection with spectral spaces

Priestley spaces are closely related to spectral spaces. For a Priestley space (X,τ,≤), let τu denote the collection of allopen up-sets of X. Similarly, let τd denote the collection of all open down-sets of X.Theorem:[4] If (X,τ,≤) is a Priestley space, then both (X,τu) and (X,τd) are spectral spaces.Conversely, given a spectral space (X,τ), let τ# denote the patch topology on X; that is, the topology generated by thesubbasis consisting of compact open subsets of (X,τ) and their complements. Let also ≤ denote the specializationorder of (X,τ).Theorem:[5] If (X,τ) is a spectral space, then (X,τ#,≤) is a Priestley space.In fact, this correspondence between Priestley spaces and spectral spaces is functorial and yields an isomorphismbetween Pries and the category Spec of spectral spaces and spectral maps.

56.4 Connection with bitopological spaces

Priestley spaces are also closely related to bitopological spaces.Theorem:[6] If (X,τ,≤) is a Priestley space, then (X,τu,τd) is a pairwise Stone space. Conversely, if (X,τ1,τ2) is apairwise Stone space, then (X,τ,≤) is a Priestley space, where τ is the join of τ1 and τ2 and ≤ is the specializationorder of (X,τ1).The correspondence between Priestley spaces and pairwise Stone spaces is functorial and yields an isomorphismbetween the category Pries of Priestley spaces and Priestley morphisms and the category PStone of pairwise Stonespaces and bi-continuous maps.Thus, one has the following isomorphisms of categories:

Spec ∼= Pries ∼= PStone

One of the main consequences of the duality theory for distributive lattices is that each of these categories is duallyequivalent to the category of bounded distributive lattices.

56.5 See also

• Spectral space

• Pairwise Stone space

• Distributive lattice

• Stone duality

• Duality theory for distributive lattices

136 CHAPTER 56. PRIESTLEY SPACE

56.6 Notes[1] Priestley, (1970).

[2] Cornish, (1975).

[3] Bezhanishvili et al. (2010)

[4] Cornish, (1975). Bezhanishvili et al. (2010).

[5] Cornish, (1975). Bezhanishvili et al. (2010).

[6] Bezhanishvili et al. (2010).

56.7 References• Priestley, H. A. (1970). Representation of distributive lattices by means of ordered Stone spaces.Bull. London

Math. Soc., (2) 186–190.

• Priestley, H. A. (1972). Ordered topological spaces and the representation of distributive lattices. Proc. LondonMath. Soc., 24(3) 507–530.

• Cornish, W. H. (1975). On H. Priestley’s dual of the category of bounded distributive lattices. Mat. Vesnik,12(27) (4) 329–332.

• M. Hochster (1969). Prime ideal structure in commutative rings. Trans. Amer. Math. Soc., 142 43–60

• Bezhanishvili, G., Bezhanishvili, N., Gabelaia, D., Kurz, A. (2010). Bitopological duality for distributivelattices and Heyting algebras. Mathematical Structures in Computer Science, 20.

Chapter 57

Projectively extended real line

In real analysis, the projectively extended real line (also called the one-point compactification of the real line, orsimply real projective line), is the extension of the number line by a point denoted ∞. It is thus the set R ∪ ∞(where R is the set of the real numbers), sometimes denoted by R. The added point is called the point at infinity,because it is considered as a neighbour of both ends of the real line. More precisely, the point at infinity is the limitof every sequence of real numbers whose absolute values are increasing and unbounded.The projectively extended real line may be identified with the projective line over the reals in which three specificpoints (e.g. 0, 1 and∞) have been chosen. The projectively extended real line must not be confused with the extendedreal number line, in which +∞ and −∞ are distinct.

57.1 Dividing by zero

Unlike most mathematical models of the intuitive concept of 'number', this structure allows division by zero:

a

0= ∞

for nonzero a. In particular 1/0 = ∞, and moreover 1/∞ = 0, making reciprocal, 1/x, a total function in this structure.The structure, however, is not a field, and division does not retain its original algebraic meaning in it, as witnessedfor example by 0⋅∞ being undefined despite reciprocation being total. The geometric interpretation is this: a verticalline has infinite slope.

57.2 Extensions of the real line

The real projective line extends the field of real numbers in the same way that the Riemann sphere extends the fieldof complex numbers, by adding a single point called conventionally∞ .Compare with the extended real number line (also called the two-point compactification of the real line), whichdistinguishes between +∞ and −∞ .

57.3 Order

The order relation cannot be extended to R in a meaningful way. Given a real number a, there is no convincing reasonto decide that a > ∞ or that a < ∞ . Since ∞ can't be compared with any of the other elements, there’s no pointin using this relation at all. However, order is used to make definitions in R that are based on the properties of reals.

137

138 CHAPTER 57. PROJECTIVELY EXTENDED REAL LINE

∞

0 The real projective line can be thought of as a line whose endpoints meet at infinity.

57.4 Geometry

Fundamental to the idea that∞ is a point no different from any other is the way the real projective line is a homogeneousspace, in fact homeomorphic to a circle. For example the general linear group of 2×2 real invertible matrices has atransitive action on it. The group action may be expressed by Möbius transformations, (also called linear fractionaltransformations), with the understanding that when the denominator of the linear fractional transformation is 0, theimage is ∞.The detailed analysis of the action shows that for any three distinct points P, Q and R, there is a linear fractionaltransformation taking P to 0, Q to 1, andR to∞ that is, the group of linear fractional transformations is triply transitiveon the real projective line. This cannot be extended to 4-tuples of points, because the cross-ratio is invariant.

57.5. ARITHMETIC OPERATIONS 139

The terminology projective line is appropriate, because the points are in 1-to-1 correspondence with one-dimensionallinear subspaces of R2.

57.5 Arithmetic operations

57.5.1 Motivation for arithmetic operations

The arithmetic operations on this space are an extension of the same operations on reals. A motivation for the newdefinitions is the limits of functions of real numbers.

57.5.2 Arithmetic operations that are defined

a+∞ = ∞+ a = ∞, a ∈ Ra−∞ = ∞− a = ∞, a ∈ Ra · ∞ = ∞ · a = ∞, a ∈ R, a = 0

∞ ·∞ = ∞a

∞= 0, a ∈ R

∞a

= ∞, a ∈ Ra

0= ∞, a ∈ R, a = 0

57.5.3 Arithmetic operations that are left undefined

The following cannot be motivated by considering limits of real functions, and any definition of them would requireus to give up additional algebraic properties. Therefore, they are left undefined:

∞+∞∞−∞∞ · 00 · ∞∞∞0

0

57.6 Algebraic properties

The following equalities mean: Either both sides are undefined, or both sides are defined and equal. This is true forany a, b, c ∈ R .

(a+ b) + c = a+ (b+ c)

a+ b = b+ a

(a · b) · c = a · (b · c)a · b = b · a

a · ∞ =a

0

The following is true whenever the right-hand side is defined, for any a, b, c ∈ R .

140 CHAPTER 57. PROJECTIVELY EXTENDED REAL LINE

a · (b+ c) = a · b+ a · c

a = (a

b) · b =

(a · b)b

a = (a+ b)− b = (a− b) + b

In general, all laws of arithmetic are valid as long as all the occurring expressions are defined.

57.7 Intervals and topology

The concept of an interval can be extended to R . However, since it is an unordered set, the interval has a slightlydifferent meaning. The definitions for closed intervals are as follows (it is assumed that a, b ∈ R, a < b ):

[a, a] = a[a, b] = x | x ∈ R, a ≤ x ≤ b

[a,∞] = x | x ∈ R, a ≤ x ∪ ∞[b, a] = x | x ∈ R, b ≤ x ∪ ∞ ∪ x | x ∈ R, x ≤ a

[∞, a] = ∞ ∪ x | x ∈ R, x ≤ a[∞,∞] = ∞

The corresponding open and half-open intervals are obtained by removing the respective endpoints.R itself is also an interval, as is R excluding any single point, but these cannot be represented with this bracketnotation.The open intervals as base define a topology on R . Sufficient for a base are the finite open intervals and the intervals(b, a) = x | x ∈ R, b < x ∪ ∞ ∪ x | x ∈ R, x < a .As said, the topology is homeomorphic to a circle. Thus it is metrizable corresponding (for a given homeomorphism)to the ordinary metric on this circle (either measured straight or along the circle). There is no metric which is anextension of the ordinary metric on R.

57.8 Interval arithmetic

Interval arithmetic is trickier in R than in R . However, the result of an arithmetic operation on intervals is always aninterval. In particular, we have, for every a, b ∈ R :

x ∈ [a, b] ⇐⇒ 1

x∈[1

b,1

a

]which is true even when the intervals involved include 0.

57.9 Calculus

The tools of calculus can be used to analyze functions of R . The definitions are motivated by the topology of thisspace.

57.9.1 Neighbourhoods

Let x ∈ R, A ⊆ R .

57.9. CALCULUS 141

• A is a neighbourhood of x, if and only if A contains an open interval B and x ∈ B .

• A is a right-sided neighbourhood of x, if and only if there is y ∈ R, y > x such that A contains [x, y) .

• A is a left-sided neighbourhood of x, if and only if there is y ∈ R, y < x such that A contains (y, x] .

• A is a (right-sided, left-sided) punctured neighbourhood of x, if and only if there is B ⊆ R such that B is a(right-sided, left-sided) neighbourhood of x, and A = B \ x .

57.9.2 Limits

Basic definitions of limits

Let f : R → R, p ∈ R, L ∈ R .The limit of f(x) as x approaches p is L, denoted

limx→p

f(x) = L

if and only if for every neighbourhood A of L, there is a punctured neighbourhood B of p, such that x ∈ B impliesf(x) ∈ A .The one-sided limit of f(x) as x approaches p from the right (left) is L, denoted

limx→p+ f(x) = L(limx→p− f(x) = L

)if and only if for every neighbourhood A of L, there is a right-sided (left-sided) punctured neighbourhood B of p, suchthat x ∈ B implies f(x) ∈ A .It can be shown that limx→p f(x) = L if and only if both limx→p+ f(x) = L and limx→p− f(x) = L .

Comparison with limits in R

The definitions given above can be compared with the usual definitions of limits of real functions. In the followingstatements, p, L ∈ R , the first limit is as defined above, and the second limit is in the usual sense:

• limx→p f(x) = L is equivalent to limx→p f(x) = L .

• limx→∞+ f(x) = L is equivalent to limx→−∞ f(x) = L .

• limx→∞− f(x) = L is equivalent to limx→+∞ f(x) = L .

• limx→p f(x) = ∞ is equivalent to limx→p |f(x)| = +∞ .

• limx→∞+ f(x) = ∞ is equivalent to limx→−∞ |f(x)| = +∞ .

• limx→∞− f(x) = ∞ is equivalent to limx→+∞ |f(x)| = +∞ .

Extended definition of limits

Let A ⊆ R . Then p is a limit point of A if and only if every neighbourhood of p includes a point y ∈ A such thaty = x .Let f : R → R, A ⊆ R, L ∈ R, p ∈ R , p a limit point of A. The limit of f(x) as x approaches p through A is L, ifand only if for every neighbourhood B of L, there is a punctured neighbourhood C of p, such that x ∈ A∩C impliesf(x) ∈ B .This corresponds to the regular topological definition of continuity, applied to the subspace topology on A ∪ p ,and the restriction of f to A ∪ p .

142 CHAPTER 57. PROJECTIVELY EXTENDED REAL LINE

57.9.3 Continuity

Let

f : R → R, p ∈ R.

f is continuous at p if and only if f is defined at p and:

limx→p

f(x) = f(p).

Let

f : R → R, A ⊆ R.

f is continuous in A if and only if for every p ∈ A , f is defined at p and the limit of f(x) as x approaches p throughA is f(p).An interesting feature is that every rational function P(x)/Q(x), where P(x) and Q(x) have no common factor, iscontinuous in R . Also, If tan is extended so that

tan(π2+ nπ

)= ∞ for n ∈ Z,

then tan is continuous in R . However, many elementary functions, such as trigonometric and exponential functions,are discontinuous at∞ . For example, sin is continuous in R but discontinuous at∞ .Thus 1/x is continuous on R but not on the affinely extended real number system R. Conversely, the function arctancan be extended continuously on R, but not on R .

57.10 As a projective range

Main article: Projective range

When the real projective line is considered in the context of the real projective plane, then the consequences ofDesargues’ theorem are implicit. In particular, the construction of the projective harmonic conjugate relation betweenpoints is part of the structure of the real projective line. For instance, given any pair of points, the point at infinity isthe projective harmonic conjugate of their midpoint.As projectivities preserve the harmonic relation, they form the automorphisms of the real projective line. The pro-jectivities are described algebraically as homographies, since the real numbers form a ring, according to the generalconstruction of a projective line over a ring. Collectively they form the group PGL(2,R).The projectivities which are their own inverses are called involutions. A hyperbolic involution has two fixed points.Two of these correspond to elementary, arithmetic operations on the real projective line: negation and reciprocation.Indeed, 0 and ∞ are fixed under negation, while 1 and −1 are fixed under reciprocation.

57.11 See also• Projective line

• Real projective plane

• Complex projective plane

• Wheel theory

• Point at infinity

57.12. EXTERNAL LINKS 143

57.12 External links• Projectively Extended Real Numbers -- From Mathworld

Chapter 58

Prüfer manifold

In mathematics, the Prüfer manifold or Prüfer surface is a 2-dimensional Hausdorff real analytic manifold that isnot paracompact. It was introduced by Radó (1925) and named after Heinz Prüfer.

58.1 Construction

The Prüfer manifold can be constructed as follows (Spivak 1979, appendix A). Take an uncountable number of copiesXa of the plane, one for each real number a, and take a copy H of the upper half plane (of pairs (x, y) with y > 0).Then glue each plane Xa to the upper half plane H by identifying (x,y)∈Xa for y > 0 with the point (a + yx, y) inH. The resulting quotient space is the Prüfer manifold. The images of the points (0,0) in the spaces Xa form anuncountable discrete subset.

58.2 See also• Long line (topology)

58.3 References• Radó, T. (1925), "Über den Begriff der Riemannschen Flächen”, Acta Litt. Sci. Szeged 2: 101–121

• Solomentsev, E.D. (2001), “Prüfer surface”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

• Spivak, Michael (1979), A comprehensive introduction to differential geometry. Vol. I (2nd ed.), Houston, TX:Publish or Perish, ISBN 978-0-914098-83-6, MR 532830

144

Chapter 59

Pseudocircle

The pseudocircle is the finite topological space X consisting of four distinct points a,b,c,d with the followingnon-Hausdorff topology:

a, b, c, d, a, b, c, a, b, d, a, b, a, b, ∅ . This topology corresponds to the partial ordera < c, b < c, a < d, b < d where open sets are downward closed sets.

X is highly pathological from the usual viewpoint of general topology as it fails to satisfy any separation axiom besidesT0. However, from the viewpoint of algebraic topologyX has the remarkable property that it is indistinguishable fromthe circle S1.More precisely the continuous map f from S1 to X (where we think of S1 as the unit circle in R2) given by

f(x, y) =

a x < 0

b x > 0

c (x, y) = (0, 1)

d (x, y) = (0,−1)

is a weak homotopy equivalence, that is f induces an isomorphism on all homotopy groups. It follows (proposition4.21 in Hatcher) that f also induces an isomorphism on singular homology and cohomology and more generally anisomorphism on all ordinary or extraordinary homology and cohomology theories (e.g., K-theory).This can be proved using the following observation. Like S1, X is the union of two contractible open sets a,b,c anda,b,d whose intersection a,b is also the union of two disjoint contractible open sets a and b. So like S1, theresult follows from the groupoid Seifert-van Kampen Theorem, as in the book “Topology and Groupoids”.More generally McCord has shown that for any finite simplicial complex K, there is a finite topological space XKwhich has the same weak homotopy type as the geometric realization |K | of K. More precisely there is a functor,taking K to XK, from the category of finite simplicial complexes and simplicial maps and a natural weak homotopyequivalence from |K | to XK.

59.1 References1. Michael C. McCord (1966). “Singular homology groups and homotopy groups of finite topological spaces”.

Duke Mathematical Journal 33: 465–474. doi:10.1215/S0012-7094-66-03352-7.

2. Algebraic Topology, by Allen Hatcher, Cambridge University Press, 2002.

1. Ronald Brown, “Topology and Groupoids”, Bookforce (2006). Available from amazon.

145

Chapter 60

Pseudomanifold

A pseudomanifold is a special type of topological space. It looks like a manifold at most of the points, but maycontain singularities. For example, the cone of solutions of z2 = x2 + y2 forms a pseudomanifold.

A pinched torus

A pseudomanifold can be regarded as a combinatorial realisation of the general idea of a manifold with singularities.The concepts of orientability, orientation and degree of a mapping make sense for pseudomanifolds and moreover,within the combinatorial approach, pseudomanifolds form the natural domain of definition for these concepts.[1][2]

60.1 Definition

A topological space X endowed with a triangulation K is an n-dimensional pseudomanifold if the following conditionshold:[3]

1. (pure) X = |K | is the union of all n-simplices.

146

60.2. RELATED DEFINITIONS 147

2. Every (n – 1)-simplex is a face of exactly two n-simplices for n > 1.

3. For every pair of n-simplices σ and σ' in K, there is a sequence of n-simplices σ = σ0, σ1, …, σk = σ' such thatthe intersection σi ∩ σi₊₁ is an (n − 1)-simplex for all i.

60.1.1 Implications of the definition

• Condition 2 means that X is a non-branching simplicial complex.[4]

• Condition 3 means that X is a strongly connected simplicial complex.[4]

60.2 Related definitions• A pseudomanifold is called normal if link of each simplex with codimension ≥ 2 is a pseudomanifold.

60.3 Examples• A pinched torus (see figure) is an example of an orientable, compact 2-dimensional pseudomanifold.[3]

• Complex algebraic varieties (even with singularities) are examples of pseudomanifolds.[4]

• Thom spaces of vector bundles over triangulable compact manifolds are examples of pseudomanifolds.[4]

• Triangulable, compact, connected, homology manifolds over Z are examples of pseudomanifolds.[4]

60.4 References[1] Steifert, H.; Threlfall, W. (1980), Textbook of Topology, Academic Press Inc., ISBN 0-12-634850-2

[2] Spanier, H. (1966), Algebraic Topology, McGraw-Hill Education, ISBN 0-07-059883-5

[3] Brasselet, J. P. (1996). “Intersection of Algebraic Cycles”. Journal of Mathematical Sciences (Springer New York) 82 (5):3625 − 3632. doi:10.1007/bf02362566.

[4] D. V. Anosov. “Pseudo-manifold”. Retrieved August 6, 2010.

Chapter 61

Ran space

In mathematics, the Ran space (or Ran’s space) of a topological space X is a topological space Ran(X) whoseunderlying set is the set of all nonempty finite subsets of X: for a metric space X the topology is given by Hausdorffdistance. The notion is named after Ziv Ran. It seems the notion was first introduced and popularized by A. Beilinsonand V. Drinfeld, Chiral algebras.In general, the topology of the Ran space is generated by sets

S ∈ Ran(U1 ∪ · · · ∪ Um) | S ∩ U1 = ∅, . . . , S ∩ Um = ∅

for any disjoint open subsets Ui ⊂ X, 1 ≤ i ≤ m .A theorem of Beilinson and Drinfeld states that the Ran space of a connected manifold is weakly contractible.[1]

There is an analog of a Ran space for a scheme:[2] the Ran prestack of a quasi-projective scheme X over a field k,denoted by Ran(X) , is the category where the objects are triples (R,S, µ) consisting of a finitely generated k-algebraR, a nonempty set S and a map of sets µ : S → X(R) and where a morphism (R,S, µ) → (R′, S′, µ′) consists ofa k-algebra homomorphism R → R′ , a surjective map S → S′ that commutes with µ and µ′ . Roughly, an R-pointof Ran(X) is a nonempty finite set of R-rational points of X “with labels” given by µ . A theorem of Beilinson andDrinfeld continues to hold: Ran(X) is acyclic if X is connected.

61.1 Topological chiral homology

If F is a cosheaf on the Ran space Ran(M) , then its space of global sections is called the topological chiral homologyofM with coefficients in F. If A is, roughly, a family of commutative algebras parametrized by points inM, then thereis a factorizable sheaf associated to A. Via this construction, one also obtains the topological chiral homology withcoefficients in A. The construction is a generalization of Hochschild homology.[3]

61.2 See also• Chiral homology

• Exponential space

61.3 Notes[1] Lurie 2012, Theorem 5.3.1.6.

[2] http://www.math.harvard.edu/~lurie/282ynotes/LectureVII-Stacks.pdf

[3] Lurie 2012, Theorem 5.3.3.11

148

61.4. REFERENCES 149

61.4 References• D. Gaitsgory, Contractibility of the space of rational maps, 2012

• http://www.math.harvard.edu/~lurie/282ynotes/LectureVIII-Poincare.pdf

• J. Lurie, Higher Algebra, last updated August 2012

• http://pantodon.shinshu-u.ac.jp/topology/literature/ja/exponential_space.html

Chapter 62

Real line

This article covers advanced topics. For basic topics, see Number line.In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real

The real line

line is the set R of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one. Itcan be thought of as a vector space (or affine space), a metric space, a topological space, a measure space, or a linearcontinuum.Just like the set of real numbers, the real line is usually denoted by the symbol R (or alternatively, R , the letter “R”in blackboard bold). However, it is sometimes denoted R1 in order to emphasize its role as the first Euclidean space.This article focuses on the aspects of R as a geometric space in topology, geometry, and real analysis. The realnumbers also play an important role in algebra as a field, but in this context R is rarely referred to as a line. For moreinformation on R in all of its guises, see real number.

62.1 As a linear continuum

0 1 2 3-1-2-3

x y<

The order on the number line

The real line is a linear continuum under the standard < ordering. Specifically, the real line is linearly ordered by <,and this ordering is dense and has the least-upper-bound property.

150

62.2. AS A METRIC SPACE 151

set M

upper bounds of Msupremum = leastupper bound

Each set on the real number line has a supremum.

In addition to the above properties, the real line has no maximum or minimum element. It also has a countable densesubset, namely the set of rational numbers. It is a theorem that any linear continuum with a countable dense subsetand no maximum or minimum element is order-isomorphic to the real line.The real line also satisfies the countable chain condition: every collection ofmutually disjoint, nonempty open intervalsin R is countable. In order theory, the famous Suslin problem asks whether every linear continuum satisfying thecountable chain condition that has no maximum or minimum element is necessarily order-isomorphic to R. Thisstatement has been shown to be independent of the standard axiomatic system of set theory known as ZFC.

62.2 As a metric space

The metric on the real line is absolute difference.

aa-ε a+ε

An ε-ball around a number a.

The real line forms a metric space, with the distance function given by absolute difference:

d(x, y) = | x − y | .

The metric tensor is clearly the 1-dimensional Euclidean metric. Since the n-dimensional Euclidean metric can berepresented in matrix form as the n by n identity matrix, the metric on the real line is simply the 1 by 1 identitymatrix, i.e. 1.If p ∈ R and ε > 0, then the ε-ball in R centered at p is simply the open interval (p − ε, p + ε).This real line has several important properties as a metric space:

• The real line is a complete metric space, in the sense that any Cauchy sequence of points converges.

152 CHAPTER 62. REAL LINE

• The real line is path-connected, and is one of the simplest examples of a geodesic metric space

• The Hausdorff dimension of the real line is equal to one.

62.3 As a topological space

∞

0 The real line can be compactified by adding a point at infinity.

The real line carries a standard topology which can be introduced in two different, equivalent ways. First, since thereal numbers are totally ordered, they carry an order topology. Second, the real numbers inherit a metric topologyfrom the metric defined above. The order topology and metric topology on R are the same. As a topological space,the real line is homeomorphic to the open interval (0, 1).

62.4. AS A VECTOR SPACE 153

The real line is trivially a topological manifold of dimension 1. Up to homeomorphism, it is one of only two different1-manifolds without boundary, the other being the circle. It also has a standard differentiable structure on it, makingit a differentiable manifold. (Up to diffeomorphism, there is only one differentiable structure that the topologicalspace supports.)The real line is locally compact and paracompact, as well as second-countable and normal. It is also path-connected,and is therefore connected as well, though it can be disconnected by removing any one point. The real line is alsocontractible, and as such all of its homotopy groups and reduced homology groups are zero.As a locally compact space, the real line can be compactified in several different ways. The one-point compactificationof R is a circle (namely the real projective line), and the extra point can be thought of as an unsigned infinity.Alternatively, the real line has two ends, and the resulting end compactification is the extended real line [−∞, +∞].There is also the Stone–Čech compactification of the real line, which involves adding an infinite number of additionalpoints.In some contexts, it is helpful to place other topologies on the set of real numbers, such as the lower limit topologyor the Zariski topology. For the real numbers, the latter is the same as the finite complement topology.

62.4 As a vector space

0

x

x

The bijection between points on the real line and vectors.

The real line is a vector space over the field R of real numbers (that is, over itself) of dimension 1. It has a standardinner product, making it a Euclidean space. (The inner product is simply ordinary multiplication of real numbers.)The standard norm on R is simply the absolute value function.

62.5 As a measure space

The real line carries a canonical measure, namely the Lebesgue measure. This measure can be defined as thecompletion of a Borel measure defined on R, where the measure of any interval is the length of the interval.Lebesgue measure on the real line is one of the simplest examples of a Haar measure on a locally compact group.

62.6 In real algebras

The real line is a one-dimensional subspace of a real algebra A where R ⊂ A. For example, in the complex plane z =x + iy, the subspace z : y = 0 is a real line. Similarly, the algebra of quaternions

q = w + x i + y j + z k

has a real line in the subspace q : x = y = z = 0 .When the real algebra is a direct sum A = R ⊕ V, then a conjugation on A is introduced by the mapping v 7→ −vof subspace V. In this way the real line consists of the fixed points of the conjugation.

154 CHAPTER 62. REAL LINE

62.7 See also• Line (geometry)

• Imaginary line (mathematics)

• Real projective line

62.8 References• Munkres, James (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.

• Walter Rudin, Real and Complex Analysis, McGraw-Hill, 1966, ISBN 0-07-100276-6.

Chapter 63

Rose (topology)

A rose with four petals.

In mathematics, a rose (also known as a bouquet of n circles) is a topological space obtained by gluing together acollection of circles along a single point. The circles of the rose are called petals. Roses are important in algebraictopology, where they are closely related to free groups.

155

156 CHAPTER 63. ROSE (TOPOLOGY)

63.1 Definition

The fundamental group of the figure eight is the free group generated by a and b

A rose is a wedge sum of circles. That is, the rose is the quotient space C/S, where C is a disjoint union of circles andS a set consisting of one point from each circle. As a cell complex, a rose has a single vertex, and one edge for eachcircle. This makes it a simple example of a topological graph.A rose with n petals can also be obtained by identifying n points on a single circle. The rose with two petals is knownas the figure eight.

63.2 Relation to free groups

The fundamental group of a rose is free, with one generator for each petal. The universal cover is an infinite tree,which can be identified with the Cayley graph of the free group. (This is a special case of the presentation complexassociated to any presentation of a group.)The intermediate covers of the rose correspond to subgroups of the free group. The observation that any cover of arose is a graph provides a simple proof that every subgroup of a free group is free (the Nielsen–Schreier theorem)Because the universal cover of a rose is contractible, the rose is actually an Eilenberg–MacLane space for the asso-ciated free group F. This implies that the cohomology groups Hn(F) are trivial for n ≥ 2.

63.3 Other properties

• Any connected graph is homotopy equivalent to a rose. Specifically, the rose is the quotient space of the graphobtained by collapsing a spanning tree.

• A disc with n points removed (or a sphere with n + 1 points removed) deformation retracts onto a rose with npetals. One petal of the rose surrounds each of the removed points.

• A torus with one point removed deformation retracts onto a figure eight, namely the union of two generatingcircles. More generally, a surface of genus g with one point removed deformation retracts onto a rose with 2gpetals, namely the boundary of a fundamental polygon.

• A rose can have infinitely many petals, leading to a fundamental group which is free on infinitely many gener-ators. The rose with countably infinitely many petals is similar to the Hawaiian earring: there is a continuousbijection from this rose onto the Hawaiian earring, but the two are not homeomorphic.

63.4 See also

• Quadrifolium

63.5. REFERENCES 157

e a

b

The universal cover of the figure eight can be visualized by the Cayley graph of the free group on two generators a and b

• Free group

• Topological graph

• Hawaiian earring

63.5 References• Hatcher, Allen (2002),Algebraic topology, Cambridge, UK: Cambridge University Press, ISBN 0-521-79540-0

• Munkres, James R. (2000), Topology, Englewood Cliffs, N.J: Prentice Hall, Inc, ISBN 0-13-181629-2

• Stillwell, John (1993), Classical topology and combinatorial group theory, Berlin: Springer-Verlag, ISBN 0-387-97970-0

158 CHAPTER 63. ROSE (TOPOLOGY)

A figure eight in the torus.

Chapter 64

Shrinking space

In mathematics, in the field of topology, a topological space is said to be a shrinking space if every open coveradmits a shrinking. A shrinking of an open cover is another open cover indexed by the same indexing set, with theproperty that the closure of each open set in the shrinking lies inside the corresponding original open set.[1]

The following facts are known about shrinking spaces:

• Every shrinking space is normal.[1]

• Every shrinking space is countably paracompact.[1]

• In a normal space, every locally finite, and in fact, every point finite open cover admits a shrinking.[1]

• Thus, every normal metacompact space is a shrinking space. In particular, every paracompact space is ashrinking space.[1]

These facts are particularly important because shrinking of open covers is a common technique in the theory ofdifferential manifolds and while constructing functions using a partition of unity.

64.1 References[1] Hart, K. P.; Nagata, Jun-iti; Vaughan, J. E. (2003), Encyclopedia ofGeneral Topology, Elsevier, p. 199, ISBN9780080530864.

• General topology, Stephen Willard, definition 15.9 p.104

159

Chapter 65

Sierpinski carpet

6 steps of the Sierpinski carpet.

The Sierpinski carpet is a plane fractal first described byWacław Sierpiński in 1916. The carpet is one generalizationof the Cantor set to two dimensions; another is the Cantor dust.The technique of subdividing a shape into smaller copies of itself, removing one or more copies, and continuing

160

65.1. CONSTRUCTION 161

recursively can be extended to other shapes. For instance, subdividing an equilateral triangle into four equilateraltriangles, removing the middle triangle, and recursing leads to the Sierpinski triangle. In three dimensions, a similarconstruction based on cubes produces the Menger sponge.

65.1 Construction

The construction of the Sierpinski carpet begins with a square. The square is cut into 9 congruent subsquares in a3-by-3 grid, and the central subsquare is removed. The same procedure is then applied recursively to the remaining8 subsquares, ad infinitum. It can be realised as the set of points in the unit square whose coordinates written in basethree do not both have a digit '1' in the same position.[1]

The process of recursively removing squares is an example of a finite subdivision rule.The Sierpinski carpet can also be created by iterating every pixel in a square and using the following algorithm todecide if the pixel is filled. The following implementation is valid C, C++, and Java./** * Decides if a point at a specific location is filled or not. This works by iteration first checking if * the pixel isunfilled in successively larger squares or cannot be in the center of any larger square. * @param x is the x coordinateof the point being checked with zero being the first pixel * @param y is the y coordinate of the point being checkedwith zero being the first pixel * @return 1 if it is to be filled or 0 if it is open */ int isSierpinskiCarpetPixelFilled(intx, int y) while(x>0 || y>0) // when either of these reaches zero the pixel is determined to be on the edge // at thatsquare level and must be filled if(x%3==1 && y%3==1) //checks if the pixel is in the center for the current squarelevel return 0; x /= 3; //x and y are decremented to check the next larger square level y /= 3; return 1; // if all possiblesquare levels are checked and the pixel is not determined // to be open it must be filled

65.1.1 Process

65.2 Properties

The area of the carpet is zero (in standard Lebesgue measure). Proof: Denote by ai the area of iteration i. Thenai₊₁=8⁄9⋅ai. So ai=(8⁄9)i, which tends to 0 as i goes to infinity.The interior of the carpet is empty. Proof: Suppose by contradiction that there is a point P in the interior of thecarpet. Then there is a square centered at P which is entirely contained in the carpet. This square contains a smallersquare whose coordinates are multiples of 1⁄₃ for some k. But, this square must have been holed in iteration k, so itcan't be contained in the carpet - a contradiction.The Hausdorff dimension of the carpet is log 8/log 3 ≈ 1.8928.[2]

Sierpiński demonstrated that his carpet is a universal plane curve.[3] That is: the Sierpinski carpet is a compact subset

162 CHAPTER 65. SIERPINSKI CARPET

of the plane with Lebesgue covering dimension 1, and every subset of the plane with these properties is homeomorphicto some subset of the Sierpinski carpet.This 'universality' of the Sierpinski carpet is not a universal property in the sense of category theory: it does notuniquely characterize this space up to homeomorphism. For example, the disjoint union of a Sierpinski carpet anda circle is also a universal plane curve. However, in 1958 Gordon Whyburn[4] uniquely characterized the Sierpinskicarpet as follows: any curve that is locally connected and has no 'local cut-points’ is homeomorphic to the Sierpinskicarpet. Here a local cut-point is a point p for which some connected neighborhood U of p has the property that U -p is not connected. So, for example, any point of the circle is a local cut point.In the same paper Whyburn gave another characterization of the Sierpinski carpet. Recall that a continuum is anonempty connected compact metric space. Suppose X is a continuum embedded in the plane. Suppose its comple-ment in the plane has countably many connected components C1, C2, C3, . . . and suppose:

• the diameter of Ci goes to zero as i → ∞ ;

• the boundary of Ci and the boundary of Cj are disjoint if i = j ;

• the boundary of Ci is a simple closed curve for each i ;

• the union of the boundaries of the sets Ci is dense in X.

Then X is homeomorphic to the Sierpinski carpet.

65.3 Brownian motion on the Sierpinski carpet

The topic of Brownian motion on the Sierpinski carpet has attracted interest in recent years.[5] Martin Barlow andRichard Bass have shown that a random walk on the Sierpinski carpet diffuses at a slower rate than an unrestrictedrandom walk in the plane. The latter reaches a mean distance proportional to n1/2 after n steps, but the random walkon the discrete Sierpinski carpet reaches only a mean distance proportional to n1/β for some β > 2. They also showedthat this random walk satisfies stronger large deviation inequalities (so called “sub-gaussian inequalities”) and that itsatisfies the elliptic Harnack inequality without satisfying the parabolic one. The existence of such an example wasan open problem for many years.

65.4 Wallis sieve

A variation of the Sierpinski carpet, called the Wallis sieve, starts in the same way, by subdividing the unit squareinto nine smaller squares and removing the middle of them. At the next level of subdivision, it subdivides each ofthe squares into 25 smaller squares and removes the middle one, and it continues at the ith step by subdividing eachsquare into (2i + 1)2 smaller squares and removing the middle one.By the Wallis product, the area of the resulting set is π/4,[6][7] unlike the standard Sierpinski carpet which has zerolimiting area.However, by the results of Whyburn mentioned above, we can see that the Wallis sieve is homeomorphic to theSierpinski carpet. In particular, its interior is still empty.

65.5 Applications

Mobile phone and WiFi fractal antennas have been produced in the form of few iterations of the Sierpinski carpet.Due to their self-similarity and scale invariance, they easily accommodate multiple frequencies. They are also easy tofabricate and smaller than conventional antennas of similar performance, thus being optimal for pocket-sized mobilephones.

65.6. SEE ALSO 163

65.6 See also• List of fractals by Hausdorff dimension

• Hawaiian earring

65.7 References[1] Allouche, Jean-Paul; Shallit, Jeffrey (2003). Automatic Sequences: Theory, Applications, Generalizations. Cambridge

University Press. pp. 405–406. ISBN 978-0-521-82332-6. Zbl 1086.11015.

[2] Semmes, Stephen (2001). Some Novel Types of Fractal Geometry. Oxford Mathematical Monographs. Oxford UniversityPress. p. 31. ISBN 0-19-850806-9. Zbl 0970.28001.

[3] Sierpiński, Wacław (1916). “Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbedonnée”. C. r. hebd. Seanc. Acad. Sci., Paris (in French) 162: 629–632. ISSN 0001-4036. JFM 46.0295.02.

[4] Whyburn, Gordon (1958). “Topological chcracterization of the Sierpinski curve”. Fund. Math. 45: 320–324.

[5] Barlow, Martin; Bass, Richard, Brownian motion and harmonic analysis on Sierpinski carpets (PDF), retrieved 25 Septem-ber 2011

[6] Rummler, Hansklaus (1993), “Squaring the circle with holes”, The American Mathematical Monthly 100 (9): 858–860,doi:10.2307/2324662, MR 1247533.

[7] Weisstein, Eric W., “Wallis Sieve”, MathWorld.

65.8 External links• Variations on the Theme of Tremas II

• Sierpiński Cookies

• Sierpinski Carpet Project

Chapter 66

Sierpinski triangle

Sierpinski triangle

The Sierpinski triangle (also with the original orthography Sierpiński), also called the Sierpinski gasket or theSierpinski Sieve, is a fractal and attractive fixed set with the overall shape of an equilateral triangle, subdividedrecursively into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examplesof self-similar sets, i.e., it is a mathematically generated pattern that can be reproducible at any magnification orreduction. It is named after the Polish mathematician Wacław Sierpiński but appeared as a decorative pattern manycenturies prior to the work of Sierpiński.

164

66.1. CONSTRUCTIONS 165

Generated using a random algorithm

66.1 Constructions

There are many different ways of constructing the Sierpinski triangle.

66.1.1 Removing triangles

The Sierpinski triangle may be constructed from an equilateral triangle by repeated removal of triangular subsets:

1. Start with an equilateral triangle.

2. Subdivide it into four smaller congruent equilateral triangles and remove the central one.

3. Repeat step 2 with each of the remaining smaller triangles

166 CHAPTER 66. SIERPINSKI TRIANGLE

Sierpinski triangle in logic: The first 16 conjunctions of lexicographically ordered argumentsThe columns interpreted as binary numbers give 1, 3, 5, 15, 17, 51... (sequence A001317 in OEIS)

Each removed triangle (a trema) is topologically an open set.[1] This process of recursively removing triangles is anexample of a finite subdivision rule.

66.1.2 Shrinking and duplication

The same sequence of shapes, converging to the Sierpinski triangle, can alternatively be generated by the followingsteps:

1. Start with any triangle in a plane (any closed, bounded region in the plane will actually work). The canonicalSierpinski triangle uses an equilateral triangle with a base parallel to the horizontal axis (first image).

2. Shrink the triangle to ½ height and ½ width, make three copies, and position the three shrunken triangles sothat each triangle touches the two other triangles at a corner (image 2). Note the emergence of the central hole- because the three shrunken triangles can between them cover only 3/4 of the area of the original. (Holes arean important feature of Sierpinski’s triangle.)

3. Repeat step 2 with each of the smaller triangles (image 3 and so on).

66.1. CONSTRUCTIONS 167

Note that this infinite process is not dependent upon the starting shape being a triangle—it is just clearer that way.The first few steps starting, for example, from a square also tend towards a Sierpinski triangle. Michael Barnsley usedan image of a fish to illustrate this in his paper “V-variable fractals and superfractals.”[2]

The actual fractal is what would be obtained after an infinite number of iterations. More formally, one describes it interms of functions on closed sets of points. If we let da note the dilation by a factor of ½ about a point a, then theSierpinski triangle with corners a, b, and c is the fixed set of the transformation da U db U dc .This is an attractive fixed set, so that when the operation is applied to any other set repeatedly, the images convergeon the Sierpinski triangle. This is what is happening with the triangle above, but any other set would suffice.

66.1.3 Chaos game

If one takes a point and applies each of the transformations da , db , and dc to it randomly, the resulting points willbe dense in the Sierpinski triangle, so the following algorithm will again generate arbitrarily close approximations toit:[3]

Start by labeling p1, p2 and p3 as the corners of the Sierpinski triangle, and a random point v1. Set v ₊₁ = ½ ( v + pᵣ), where r is a random number 1, 2 or 3. Draw the points v1 to v∞. If the first point v1 was a point on the Sierpińskitriangle, then all the points v lie on the Sierpinski triangle. If the first point v1 to lie within the perimeter of thetriangle is not a point on the Sierpinski triangle, none of the points v will lie on the Sierpinski triangle, however theywill converge on the triangle. If v1 is outside the triangle, the only way v will land on the actual triangle, is if v ison what would be part of the triangle, if the triangle was infinitely large.Or more simply:

1. Take 3 points in a plane to form a triangle, you need not draw it.

2. Randomly select any point inside the triangle and consider that your current position.

3. Randomly select any one of the 3 vertex points.

4. Move half the distance from your current position to the selected vertex.

5. Plot the current position.

6. Repeat from step 3.

Note: This method is also called the chaos game, and is an example of an iterated function system. You can start fromany point outside or inside the triangle, and it would eventually form the Sierpinski Gasket with a few leftover points (ifthe starting point lies on the outline of the triangle, there are no leftover points). It is interesting to do this with penciland paper. A brief outline is formed after placing approximately one hundred points, and detail begins to appear aftera few hundred.

66.1.4 Arrowhead curve

Another construction for the Sierpinski triangle shows that it can be constructed as a curve in the plane. It is formedby a process of repeated modification of simpler curves, analogous to the construction of the Koch snowflake:

1. Start with a single line segment in the plane

168 CHAPTER 66. SIERPINSKI TRIANGLE

Animated creation of a Sierpinski triangle using the chaos game

2. Repeatedly replace each line segment of the curve with three shorter segments, forming 120° angles at eachjunction between two consecutive segments, with the first and last segments of the curve either parallel to theoriginal line segment or forming a 60° angle with it.

The resulting fractal curve is called the Sierpiński arrowhead curve, and its limiting shape is the Sierpinski triangle.[4]

66.1.5 Cellular automata

The Sierpinski triangle also appears in certain cellular automata (such as Rule 90), including those relating to Conway’sGame of Life. For instance, the life-like cellular automaton B1/S12 when applied to a single cell will generate fourapproximations of the Sierpinski triangle.[5] The time-space diagram of a replicator pattern in a cellular automatonalso often resembles a Sierpinski triangle.[6]

66.1.6 Pascal’s triangle

If one takes Pascal’s triangle with 2n rows and colors the even numbers white, and the odd numbers black, the resultis an approximation to the Sierpinski triangle. More precisely, the limit as n approaches infinity of this parity-colored2n-row Pascal triangle is the Sierpinski triangle.[7]

66.2. PROPERTIES 169

Animated construction of a Sierpinski triangle

66.1.7 Towers of Hanoi

The Towers of Hanoi puzzle involves moving disks of different sizes between three pegs, maintaining the propertythat no disk is ever placed on top of a smaller disk. The states of an n-disk puzzle, and the allowable moves from onestate to another, form an undirected graph that can be represented geometrically as the intersection graph of the setof triangles remaining after the nth step in the construction of the Sierpinski triangle. Thus, in the limit as n goes toinfinity, this sequence of graphs can be interpreted as a discrete analogue of the Sierpinski triangle.[8]

66.2 Properties

For integer number of dimensions d, when doubling a side of an object, 2 d copies of it are created, i.e. 2 copies for1-dimensional object, 4 copies for 2-dimensional object and 8 copies for 3-dimensional object. For Sierpinski triangledoubling its side creates 3 copies of itself. Thus Sierpinski triangle has Hausdorff dimension log(3)/log(2) ≈ 1.585,which follows from solving 2 d = 3 for d.[9]

The area of a Sierpinski triangle is zero (in Lebesgue measure). The area remaining after each iteration is clearly 3/4of the area from the previous iteration, and an infinite number of iterations results in zero.[10]

170 CHAPTER 66. SIERPINSKI TRIANGLE

Sierpinski triangle using an iterated function system

Construction of the Sierpiński arrowhead curve

The points of a Sierpinski triangle have a simple characterization in Barycentric coordinates.[11] If a point has coordi-nates (0.u1u2u3…,0.v1v2v3…,0.w1w2w3…), expressed as Binary numbers, then the point is in Sierpinski’s triangleif and only if ui+vi+wi=1 for all i.

66.3 Generalization to other Moduli

A generalization of the Sierpinski triangle can also be generated using Pascal’s Triangle if a different Modulo is used.Iteration n can be generated by taking a Pascal’s triangle with Pn rows and coloring numbers by their value for x modP. As n approaches infinity, a fractal is generated.The same fractal can be achieved by dividing a triangle into a tessellation of P2 similar triangles and removing the

66.4. ANALOGUES IN HIGHER DIMENSIONS 171

triangles that are upside-down from the original, then iterating this step with each smaller triangle.Conversely, the fractal can also be generated by beginning with a triangle and duplicating it and arranging n(n+1)/2 ofthe new figures in the same orientation into a larger similar triangle with the vertices of the previous figures touching,then iterating that step. [12]

66.4 Analogues in higher dimensions

A Sierpinski square-based pyramid and its 'inverse'

The Sierpinski tetrahedron or tetrix is the three-dimensional analogue of the Sierpinski triangle, formed by repeat-edly shrinking a regular tetrahedron to one half its original height, putting together four copies of this tetrahedronwith corners touching, and then repeating the process. This can also be done with a square pyramid and five copiesinstead. A tetrix constructed from an initial tetrahedron of side-length L has the property that the total surface arearemains constant with each iteration.The initial surface area of the (iteration-0) tetrahedron of side-length L isL2

√3 . At the next iteration, the side-length

is halved

L → L

2

and there are 4 such smaller tetrahedra. Therefore, the total surface area after the first iteration is:

4

((L

2

)2 √3

)= 4

L2

4

√3 = L2

√3.

This remains the case after each iteration. Though the surface area of each subsequent tetrahedron is 1/4 that of thetetrahedron in the previous iteration, there are 4 times as many—thus maintaining a constant total surface area.The total enclosed volume, however, is geometrically decreasing (factor of 0.5) with each iteration and asymptoticallyapproaches 0 as the number of iterations increases. In fact, it can be shown that, while having fixed area, it has no3-dimensional character. The Hausdorff dimension of such a construction is ln 4

ln 2 = 2 which agrees with the finitearea of the figure. (A Hausdorff dimension strictly between 2 and 3 would indicate 0 volume and infinite area.)

172 CHAPTER 66. SIERPINSKI TRIANGLE

A Sierpiński triangle-based pyramid as seen from above (4 main sections highlighted). Note the self-similarity in this 2-dimensionalprojected view, so that the resulting triangle could be a 2D fractal in itself.

66.5 History

Wacław Sierpiński described the Sierpinski triangle in 1915. However, similar patterns appear already in the 13th-century Cosmati mosaics in the cathedral of Anagni, Italy,[13] and other places of central Italy, for carpets in manyplaces such as the nave of the Roman Basilica of Santa Maria in Cosmedin,[14] and for isolated triangles positionedin rotae in several churches and Basiliche.[15] In the case of the isolated triangle, it is interesting to notice that theiteration is at least of three levels.

66.6 See also

• Apollonian gasket, a set of mutually tangent circles with the same combinatorial structure as the Sierpinskitriangle

• List of fractals by Hausdorff dimension

• Sierpinski carpet, another fractal named after Sierpinski and formed by repeatedly removing squares from alarger square

66.7 References[1] “Sierpinski Gasket by Trema Removal”

66.8. EXTERNAL LINKS 173

[2] Michael Barnsley; et al., V-variable fractals and superfractals (PDF)

[3] Feldman, David P. (2012), “17.4 The chaos game”, Chaos and Fractals: An Elementary Introduction, Oxford UniversityPress, pp. 178–180, ISBN 9780199566440.

[4] Prusinkiewicz, P. (1986), “Graphical applications of L−systems” (PDF), Proceedings of Graphics Interface '86 / VisionInterface '86, pp. 247–253.

[5] Rumpf, Thomas (2010), “Conway’s Game of Life accelerated with OpenCL” (PDF), Proceedings of the Eleventh Interna-tional Conference on Membrane Computing (CMC 11), pp. 459–462.

[6] Bilotta, Eleonora; Pantano, Pietro (Summer 2005), “Emergent patterning phenomena in 2D cellular automata”, ArtificialLife 11 (3): 339–362, doi:10.1162/1064546054407167.

[7] Stewart, Ian (2006), How to Cut a Cake: And other mathematical conundrums, Oxford University Press, p. 145, ISBN9780191500718.

[8] Romik, Dan (2006), “Shortest paths in the Tower of Hanoi graph and finite automata”, SIAM Journal on Discrete Mathe-matics 20 (3): 610–62, arXiv:math.CO/0310109, doi:10.1137/050628660, MR 2272218.

[9] Falconer, Kenneth (1990). Fractal geometry: mathematical foundations and applications. Chichester: John Wiley. p. 120.ISBN 0-471-92287-0. Zbl 0689.28003.

[10] Helmberg, Gilbert (2007), Getting Acquainted with Fractals, Walter de Gruyter, p. 41, ISBN 9783110190922.

[11] http://www.cut-the-knot.org/ctk/Sierpinski.shtml

[12] Shannon & Bardzell, Kathleen &Michael, “Patterns in Pascal’s Triangle - with a Twist - First Twist: What is It?", maa.org(Mathematical association of America), retrieved 29 March 2015

[13] Wolfram, Stephen (2002), A New Kind of Science, Wolfram Media, pp. 43, 873

[14] “Geometric floor mosaic (Sierpinski triangles), nave of Santa Maria in Cosmedin, Forum Boarium, Rome”, 5 September2011, Flickr

[15] Conversano, Elisa; Tedeschini-Lalli, Laura (2011), “Sierpinski Triangles in Stone on Medieval Floors in Rome"" (PDF),APLIMAT Journal of Applied Mathematics 4: 114, 122

66.8 External links• Hazewinkel, Michiel, ed. (2001), “Sierpinski gasket”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Weisstein, Eric W., “Sierpinski Sieve”, MathWorld.

• Paul W. K. Rothemund, Nick Papadakis, and Erik Winfree, Algorithmic Self-Assembly of DNA SierpinskiTriangles, PLoS Biology, volume 2, issue 12, 2004.

• Sierpinski Gasket by Trema Removal at cut-the-knot

• Sierpinski Gasket and Tower of Hanoi at cut-the-knot

• 3D printed Stage 5 Sierpinski Tetrahedron

Chapter 67

Sierpiński space

Not to be confused with Sierpiński set.

In mathematics, the Sierpiński space (or the connected two-point set) is a finite topological space with two points,only one of which is closed. It is the smallest example of a topological space which is neither trivial nor discrete. Itis named after Wacław Sierpiński.The Sierpiński space has important relations to the theory of computation and semantics.[1][2]

67.1 Definition and fundamental properties

Explicitly, the Sierpiński space is a topological space S whose underlying point set is 0,1 and whose open sets are

∅, 1, 0, 1.

The closed sets are

∅, 0, 0, 1.

So the singleton set 0 is closed (but not open) and the set 1 is open (but not closed).The closure operator on S is determined by

0 = 0, 1 = 0, 1.

A finite topological space is also uniquely determined by its specialization preorder. For the Sierpiński space thispreorder is actually a partial order and given by

0 ≤ 0, 0 ≤ 1, 1 ≤ 1.

67.2 Topological properties

The Sierpiński space S is a special case of both the finite particular point topology (with particular point 1) and thefinite excluded point topology (with excluded point 0). Therefore, S has many properties in common with one or bothof these families.

174

67.2. TOPOLOGICAL PROPERTIES 175

67.2.1 Separation

• The points 0 and 1 are topologically distinguishable in S since 1 is an open set which contains only one ofthese points. Therefore, S is a Kolmogorov (T0) space.

• However, S is not T1 since the point 1 is not closed. It follows that S is not Hausdorff, or Tn for any n ≥ 1.

• S is not regular (or completely regular) since the point 1 and the disjoint closed set 0 cannot be separated byneighborhoods. (Also regularity in the presence of T0 would imply Hausdorff.)

• S is vacuously normal and completely normal since there are no nonempty separated sets.

• S is not perfectly normal since the disjoint closed sets ∅ and 0 cannot be precisely separated by a function.Indeed, 0 cannot be the zero set of any continuous function S → R since every such function is constant.

67.2.2 Connectedness

• The Sierpiński space S is both hyperconnected (since every nonempty open set contains 1) and ultraconnected(since every nonempty closed set contains 0).

• It follows that S is both connected and path connected.

• A path from 0 to 1 in S is given by the function: f(0) = 0 and f(t) = 1 for t > 0. The function f : I → S iscontinuous since f−1(1) = (0,1] which is open in I.

• Like all finite topological spaces, S is locally path connected.

• The Sierpiński space is contractible, so the fundamental group of S is trivial (as are all the higher homotopygroups).

67.2.3 Compactness

• Like all finite topological spaces, the Sierpiński space is both compact and second-countable.

• The compact subset 1 of S is not closed showing that compact subsets of T0 spaces need not be closed.

• Every open cover of S must contain S itself since S is the only open neighborhood of 0. Therefore, every opencover of S has an open subcover consisting of a single set: S.

• It follows that S is fully normal.[3]

67.2.4 Convergence

• Every sequence in S converges to the point 0. This is because the only neighborhood of 0 is S itself.

• A sequence in S converges to 1 if and only if the sequence contains only finitely many terms equal to 0 (i.e.the sequence is eventually just 1’s).

• The point 1 is a cluster point of a sequence in S if and only if the sequence contains infinitely many 1’s.

• Examples:

• 1 is not a cluster point of (0,0,0,0,…).• 1 is a cluster point (but not a limit) of (0,1,0,1,0,1,…).• The sequence (1,1,1,1,…) converges to both 0 and 1.

67.2.5 Metrizability

• The Sierpiński space S is not metrizable or even pseudometrizable since every pseudometric space is completelyregular but the Sierpiński space it is not even regular.

• S is generated by the hemimetric (or pseudo-quasimetric) d(0, 1) = 0 and d(1, 0) = 1 .

176 CHAPTER 67. SIERPIŃSKI SPACE

67.2.6 Other properties

• There are only three continuous maps from S to itself: the identity map and the constant maps to 0 and 1.

• It follows that the homeomorphism group of S is trivial.

67.3 Continuous functions to the Sierpiński space

Let X be an arbitrary set. The set of all functions from X to the set 0,1 is typically denoted 2X. These functionsare precisely the characteristic functions of X. Each such function is of the form

χU (x) =

1 x ∈ U

0 x ∈ U

where U is a subset of X. In other words, the set of functions 2X is in bijective correspondence with P(X), the powerset of X. Every subset U of X has its characteristic function χU and every function from X to 0,1 is of this form.Now suppose X is a topological space and let 0,1 have the Sierpiński topology. Then a function χU : X → S iscontinuous if and only if χU−1(1) is open in X. But, by definition

χ−1U (1) = U.

So χU is continuous if and only if U is open in X. Let C(X,S) denote the set of all continuous maps from X to S andlet T(X) denote the topology of X (i.e. the family of all open sets). Then we have a bijection from T(X) to C(X,S)which sends the open set U to χU.

C(X,S) ∼= T (X)

That is, if we identify 2X with P(X), the subset of continuous maps C(X,S) ⊂ 2X is precisely the topology of X: T(X)⊂ P(X).

67.3.1 Categorical description

The above construction can be described nicely using the language of category theory. There is contravariant functorT : Top→ Set from the category of topological spaces to the category of sets which assigns each topological spaceX its set of open sets T(X) and each continuous function f : X → Y the preimage map

f−1 : T (Y ) → T (X).

The statement then becomes: the functor T is represented by (S, 1) where S is the Sierpiński space. That is, Tis naturally isomorphic to the Hom functor Hom(–, S) with the natural isomorphism determined by the universalelement 1 ∈ T(S).

67.3.2 The initial topology

Any topological space X has the initial topology induced by the family C(X,S) of continuous functions to Sierpińskispace. Indeed, in order to coarsen the topology on X one must remove open sets. But removing the open set U wouldrender χU discontinuous. So X has the coarsest topology for which each function in C(X,S) is continuous.The family of functions C(X,S) separates points in X if and only if X is a T0 space. Two points x and y will beseparated by the function χU if and only if the open set U contains precisely one of the two points. This is exactlywhat it means for x and y to be topologically distinguishable.

67.4. IN ALGEBRAIC GEOMETRY 177

Therefore, if X is T0, we can embed X as a subspace of a product of Sierpiński spaces, where there is one copy of Sfor each open set U in X. The embedding map

e : X →∏

U∈T (X)

S = ST (X)

is given by

e(x)U = χU (x).

Since subspaces and products of T0 spaces are T0, it follows that a topological space is T0 if and only if it ishomeomorphic to a subspace of a power of S.

67.4 In algebraic geometry

In algebraic geometry the Sierpiński space arises as the spectrum, Spec(R), of a discrete valuation ring R such as Z₍p₎(the localization of the integers at the prime ideal generated by the prime number p). The generic point of Spec(R),coming from the zero ideal, corresponds to the open point 1, while the special point of Spec(R), coming from theunique maximal ideal, corresponds to the closed point 0.

67.5 See also• Finite topological space

• Pseudocircle

67.6 Notes[1] An online paper, it explains the motivation, why the notion of “topology” can be applied in the investigation of concepts

of the computer science. Alex Simpson: Mathematical Structures for Semantics. Chapter III: Topological Spaces from aComputational Perspective. The “References” section provides many online materials on domain theory.

[2] Escardó, Martín (2004). Synthetic topology of data types and classical spaces. Electronic Notes in Theoretical ComputerScience 87. Elsevier. Retrieved July 6, 2011.

[3] Steen and Seebach incorrectly list the Sierpiński space as not being fully normal (or fully T4 in their terminology).

67.7 References• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446

Chapter 68

Simplicial complex

A simplicial 3-complex.

In mathematics, a simplicial complex is a topological space of a certain kind, constructed by “gluing together” points,line segments, triangles, and their n-dimensional counterparts (see illustration). Simplicial complexes should not beconfused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. Thepurely combinatorial counterpart to a simplicial complex is an abstract simplicial complex.

178

68.1. DEFINITIONS 179

68.1 Definitions

A simplicial complex K is a set of simplices that satisfies the following conditions:

1. Any face of a simplex from K is also in K .2. The intersection of any two simplices σ1, σ2 ∈ K is either ∅ or a face of both σ1 and σ2 .

Note that the empty set is a face of every simplex. See also the definition of an abstract simplicial complex, whichloosely speaking is a simplicial complex without an associated geometry.A simplicial k-complex K is a simplicial complex where the largest dimension of any simplex in K equals k. Forinstance, a simplicial 2-complex must contain at least one triangle, and must not contain any tetrahedra or higher-dimensional simplices.A pure or homogeneous simplicial k-complexK is a simplicial complex where every simplex of dimension less thank is a face of some simplex σ ∈ K of dimension exactly k. Informally, a pure 1-complex “looks” like it’s made of abunch of lines, a 2-complex “looks” like it’s made of a bunch of triangles, etc. An example of a non-homogeneouscomplex is a triangle with a line segment attached to one of its vertices.A facet is any simplex in a complex that is not a face of any larger simplex. (Note the difference from a “face” of asimplex). A pure simplicial complex can be thought of as a complex where all facets have the same dimension.Sometimes the term face is used to refer to a simplex of a complex, not to be confused with a face of a simplex.For a simplicial complex embedded in a k-dimensional space, the k-faces are sometimes referred to as its cells. Theterm cell is sometimes used in a broader sense to denote a set homeomorphic to a simplex, leading to the definitionof cell complex.The underlying space, sometimes called the carrier of a simplicial complex is the union of its simplices.

68.2 Closure, star, and link• Two simplices and their closure.

• A vertex and its star.

• A vertex and its link.

Let K be a simplicial complex and let S be a collection of simplices in K.The closure of S (denoted Cl S) is the smallest simplicial subcomplex of K that contains each simplex in S. Cl S isobtained by repeatedly adding to S each face of every simplex in S.The star of S (denoted St S) is the union of the stars of each simplex in S. For a single simplex s, the star of s is theset of simplices having a face in s. (Note that the star of S is generally not a simplicial complex itself).The link of S (denoted Lk S) equals Cl St S − St Cl S. It is the closed star of S minus the stars of all faces of S.

68.3 Algebraic topology

Main article: Simplicial homology

In algebraic topology, simplicial complexes are often useful for concrete calculations. For the definition of homologygroups of a simplicial complex, one can read the corresponding chain complex directly, provided that consistentorientations are made of all simplices. The requirements of homotopy theory lead to the use of more general spaces,the CW complexes. Infinite complexes are a technical tool basic in algebraic topology. See also the discussion atpolytope of simplicial complexes as subspaces of Euclidean space, made up of subsets each of which is a simplex. Thatsomewhat more concrete concept is there attributed to Alexandrov. Any finite simplicial complex in the sense talkedabout here can be embedded as a polytope in that sense, in some large number of dimensions. In algebraic topologya compact topological space which is homeomorphic to the geometric realization of a finite simplicial complex isusually called a polyhedron (see Spanier 1966, Maunder 1996, Hilton & Wylie 1967).

180 CHAPTER 68. SIMPLICIAL COMPLEX

68.4 Combinatorics

Combinatorialists often study the f-vector of a simplicial d-complexΔ, which is the integral sequence (f0, f1, f2, . . . , fd+1), where fi is the number of (i − 1)-dimensional faces of Δ (by convention, f0 = 1 unless Δ is the empty complex).For instance, if Δ is the boundary of the octahedron, then its f-vector is (1, 6, 12, 8), and if Δ is the first simplicialcomplex pictured above, its f-vector is (1, 18, 23, 8, 1). A complete characterization of the possible f-vectors ofsimplicial complexes is given by the Kruskal–Katona theorem.By using the f-vector of a simplicial d-complex Δ as coefficients of a polynomial (written in decreasing order ofexponents), we obtain the f-polynomial of Δ. In our two examples above, the f-polynomials would be x3 + 6x2 +12x+ 8 and x4 + 18x3 + 23x2 + 8x+ 1 , respectively.Combinatorists are often quite interested in the h-vector of a simplicial complex Δ, which is the sequence of coeffi-cients of the polynomial that results from plugging x − 1 into the f-polynomial of Δ. Formally, if we write FΔ(x) tomean the f-polynomial of Δ, then the h-polynomial of Δ is

F∆(x− 1) = h0xd+1 + h1x

d + h2xd−1 + · · ·+ hdx+ hd+1

and the h-vector of Δ is

(h0, h1, h2, · · · , hd+1).

We calculate the h-vector of the octahedron boundary (our first example) as follows:

F (x− 1) = (x− 1)3 + 6(x− 1)2 + 12(x− 1) + 8 = x3 + 3x2 + 3x+ 1.

So the h-vector of the boundary of the octahedron is (1, 3, 3, 1). It is not an accident this h-vector is symmetric. Infact, this happens whenever Δ is the boundary of a simplicial polytope (these are the Dehn–Sommerville equations).In general, however, the h-vector of a simplicial complex is not even necessarily positive. For instance, if we take Δto be the 2-complex given by two triangles intersecting only at a common vertex, the resulting h-vector is (1, 3, −2).A complete characterization of all simplicial polytope h-vectors is given by the celebrated g-theorem of Stanley,Billera, and Lee.Simplicial complexes can be seen to have the same geometric structure as the contact graph of a sphere packing (agraph where vertices are the centers of spheres and edges exist if the corresponding packing elements touch eachother) and as such can be used to determine the combinatorics of sphere packings, such as the number of touchingpairs (1-simplices), touching triplets (2-simplices), and touching quadruples (3-simplices) in a sphere packing.

68.5 See also• Abstract simplicial complex• Barycentric subdivision• Causal dynamical triangulation• Delta set• Polygonal chain – 1 dimensional simplicial complex• Tucker’s lemma

68.6 References• Spanier, E.H. (1966), Algebraic Topology, Springer, ISBN 0-387-94426-5• Maunder, C.R.F. (1996), Algebraic Topology, Dover, ISBN 0-486-69131-4• Hilton, P.J.; Wylie, S. (1967), Homology Theory, Cambridge University Press, ISBN 0-521-09422-4

68.7. EXTERNAL LINKS 181

68.7 External links• Weisstein, Eric W., “Simplicial complex”, MathWorld.

Chapter 69

Simplicial space

In mathematics, a simplicial space is a simplicial object in the category of topological spaces. In other words, it is acontravariant functor from the simplex category Δ to the category of topological spaces.[1]

69.1 References[1] Baues, Hans Joachim (1995), “Homotopy types”, in James, I. M., Handbook of Algebraic Topology, Amsterdam: North-

Holland, pp. 1–72, doi:10.1016/B978-044481779-2/50002-X, MR 1361886. See in particular p. 8.

182

Chapter 70

Smith–Volterra–Cantor set

After black intervals have been removed, the white points which remain are a nowhere dense set of measure 1/2.

In mathematics, the Smith–Volterra–Cantor set (SVC), fat Cantor set, or ε-Cantor set[1] is an example of a setof points on the real lineR that is nowhere dense (in particular it contains no intervals), yet has positive measure. TheSmith–Volterra–Cantor set is named after the mathematicians Henry Smith, Vito Volterra and Georg Cantor. TheSmith-Volterra-Cantor set is topologically equivalent to the middle-thirds Cantor set.

70.1 Construction

Similar to the construction of the Cantor set, the Smith–Volterra–Cantor set is constructed by removing certainintervals from the unit interval [0, 1].The process begins by removing the middle 1/4 from the interval [0, 1] (the same as removing 1/8 on either side ofthe middle point at 1/2) so the remaining set is

[0,

3

8

]∪[5

8, 1

].

The following steps consist of removing subintervals of width 1/22n from the middle of each of the 2n−1 remainingintervals. So for the second step the intervals (5/32, 7/32) and (25/32, 27/32) are removed, leaving

[0,

5

32

]∪[7

32,3

8

]∪[5

8,25

32

]∪[27

32, 1

].

Continuing indefinitely with this removal, the Smith–Volterra–Cantor set is then the set of points that are neverremoved. The image below shows the initial set and five iterations of this process.Each subsequent iterate in the Smith–Volterra–Cantor set’s construction removes proportionally less from the re-maining intervals. This stands in contrast to the Cantor set, where the proportion removed from each interval remainsconstant. Thus, the former has positive measure, while the latter zero measure.

183

184 CHAPTER 70. SMITH–VOLTERRA–CANTOR SET

70.2 Properties

By construction, the Smith–Volterra–Cantor set contains no intervals and therefore has empty interior. It is also theintersection of a sequence of closed sets, which means that it is closed. During the process, intervals of total length

∞∑n=0

2n

22n+2=

1

4+

1

8+

1

16+ · · · = 1

2

are removed from [0, 1], showing that the set of the remaining points has a positive measure of 1/2. This makes theSmith–Volterra–Cantor set an example of a closed set whose boundary has positive Lebesgue measure.

70.3 Other fat Cantor sets

In general, one can remove rn from each remaining subinterval at the n-th step of the algorithm, and end up with aCantor-like set. The resulting set will have positive measure if and only if the sum of the sequence is less than themeasure of the initial interval.Cartesian products of Smith–Volterra–Cantor sets can be used to find totally disconnected sets in higher dimensionswith nonzero measure. By applying the Denjoy–Riesz theorem to a two-dimensional set of this type, it is possible tofind a Jordan curve such that the points on the curve have positive area.[2]

70.4 See also• The SVC is used in the construction of Volterra’s function (see external link).

• The SVC is an example of a compact set that is not Jordan measurable, see Jordan measure#Extension to morecomplicated sets.

• The indicator function of the SVC is an example of a bounded function that is not Riemann integrable on(0,1) and moreover, is not equal almost everywhere to a Riemann integrable function, see Riemann inte-gral#Examples.

70.5 References[1] Aliprantis and Burkinshaw (1981), Principles of Real Analysis

[2] Balcerzak, M.; Kharazishvili, A. (1999), “On uncountable unions and intersections of measurable sets”, Georgian Mathe-matical Journal 6 (3): 201–212, doi:10.1023/A:1022102312024, MR 1679442.

70.6 External links• Wrestling with the Fundamental Theorem of Calculus: Volterra’s function, talk by David Marius Bressoud

Chapter 71

Sorgenfrey plane

An illustration of the anti-diagonal and an open rectangle in the Sorgenfrey plane that meets the anti-diagonal at a single point.

In topology, the Sorgenfrey plane is a frequently-cited counterexample to many otherwise plausible-sounding con-jectures. It consists of the product of two copies of the Sorgenfrey line, which is the real line R under the half-openinterval topology. The Sorgenfrey line and plane are named for the American mathematician Robert Sorgenfrey.

185

186 CHAPTER 71. SORGENFREY PLANE

A basis for the Sorgenfrey plane, denoted S from now on, is therefore the set of rectangles that include the westedge, southwest corner, and south edge, and omit the southeast corner, east edge, northeast corner, north edge, andnorthwest corner. Open sets in S are unions of such rectangles.S is an example of a space that is a product of Lindelöf spaces that is not itself a Lindelöf space. The so-calledanti-diagonal ∆ = (x,−x) | x ∈ R is an uncountable discrete subset of this space, and this is a non-separablesubset of the separable space S . It shows that separability does not inherit to closed subspaces. Note that K =(x,−x) | x ∈ Q and ∆ \K are closed sets that cannot be separated by open sets, showing that S is not normal.Thus it serves as a counterexample to the notion that the product of normal spaces is normal; in fact, it shows thateven the finite product of perfectly normal spaces need not be normal.

71.1 References• Kelley, John L. (1955). General Topology. van Nostrand. Reprinted as Kelley, John L. (1975). General

Topology. Springer-Verlag. ISBN 0-387-90125-6.

• Robert Sorgenfrey, “On the topological product of paracompact spaces”, Bull. Amer. Math. Soc. 53 (1947)631–632.

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978]. Counterexamples in Topology (Dover reprint of1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 507446

Chapter 72

Split interval

In topology, the split interval is a space that results from splitting each interior point in a closed interval into twoadjacent points. It may be defined as the lexicographic product [0, 1] × 0, 1 without the points the isolated edgepoints, (0,1) and (1,0), equipped with the order topology. It is also known as the Alexandrov double arrow spaceor two arrows space.The split interval is compact Hausdorff, and it is hereditarily Lindelöf and hereditarily separable, but it is notmetrizable; its metrizable subspaces are all countable.All compact, separable ordered spaces are order-isomorphic to a subset of the split interval.[1]

72.1 References[1] Ostaszewski, A. J. (February 1974), “A Characterization of Compact, Separable, Ordered Spaces”, Journal of the London

Mathematical Society s2–7 (4): 758–760, doi:10.1112/jlms/s2-7.4.758

72.2 Further reading• Todorcevic, Stevo (6 July 1999), “Compact subsets of the first Baire class”, Journal of the LondonMathematical

Society 12 (4): 1179–1212, doi:10.1090/S0894-0347-99-00312-4

187

Chapter 73

Topological monoid

In topology, a branch of mathematics, a topological monoid is a monoid object in the category of topological spaces.In other words, it is a monoid as a set and the monoid operations are continuous. A topological group is a topologicalmonoid.

73.1 See also• H-space

73.2 References• http://ncatlab.org:8080/nlab/show/topological+monoid

73.3 External links• http://mathoverflow.net/questions/114690/topological-monoid-from-symmetric-monoidal-category

188

Chapter 74

Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, alongwith a set of neighbourhoods for each point, that satisfy a set of axioms relating points and neighbourhoods. Thedefinition of a topological space relies only upon set theory and is the most general notion of a mathematical spacethat allows for the definition of concepts such as continuity, connectedness, and convergence.[1] Other spaces, such asmanifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being sogeneral, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics.The branch of mathematics that studies topological spaces in their own right is called point-set topology or generaltopology.

74.1 History of Development

74.2 Definition

Main article: Characterizations of the category of topological spaces

The utility of the notion of a topology is shown by the fact that there are several equivalent definitions of this structure.Thus one chooses the axiomatisation suited for the application. The most commonly used, and the most elegant, isthat in terms of open sets, but the most intuitive is that in terms of neighbourhoods and so we give this first. Note: Avariety of other axiomatisations of topological spaces are listed in the Exercises of the book by Vaidyanathaswamy.

74.2.1 Neighbourhoods definition

This axiomatization is due to Felix Hausdorff. Let X be a set; the elements of X are usually called points, though theycan be any mathematical object. We allow X to be empty. Let N be a function assigning to each x (point) in X anon-empty collection N(x) of subsets of X. The elements of N(x) will be called neighbourhoods of x with respect toN (or, simply, neighbourhoods of x). The function N is called a neighbourhood topology if the axioms below[2] aresatisfied; and then X with N is called a topological space.

1. If N is a neighbourhood of x (i.e., N ∈ N(x)), then x ∈ N. In other words, each point belongs to every one of itsneighbourhoods.

2. If N is a subset of X and contains a neighbourhood of x, then N is a neighbourhood of x. I.e., every supersetof a neighbourhood of a point x in X is again a neighbourhood of x.

3. The intersection of two neighbourhoods of x is a neighbourhood of x.

4. Any neighbourhood N of x contains a neighbourhood M of x such that N is a neighbourhood of each point ofM.

189

190 CHAPTER 74. TOPOLOGICAL SPACE

The first three axioms for neighbourhoods have a clear meaning. The fourth axiom has a very important use in thestructure of the theory, that of linking together the neighbourhoods of different points of X.A standard example of such a system of neighbourhoods is for the real line R, where a subset N of R is defined to bea neighbourhood of a real number x if there is an open interval containing x and contained in N.Given such a structure, we can define a subset U of X to be open if U is a neighbourhood of all points in U. It is aremarkable fact that the open sets then satisfy the elegant axioms given below, and that, given these axioms, we canrecover the neighbourhoods satisfying the above axioms by defining N to be a neighbourhood of x if N contains anopen set U such that x ∈ U.[3]

74.2.2 Open sets definition

1 2 3 1 2 3

1 2 3 1 2 3

1 2 3 1 2 3

Four examples and two non-examples of topologies on the three-point set 1,2,3. The bottom-left example is not a topology becausethe union of 2 and 3 [i.e. 2,3] is missing; the bottom-right example is not a topology because the intersection of 1,2 and2,3 [i.e. 2], is missing.

A topological space is then a set X together with a collection of subsets of X, called open sets and satisfying thefollowing axioms:[4]

1. The empty set and X itself are open.

2. Any union of open sets is open.

3. The intersection of any finite number of open sets is open.

74.3. COMPARISON OF TOPOLOGIES 191

The collection τ of open sets is then also called a topology onX, or, if more precision is needed, an open set topology.The sets in τ are called the open sets, and their complements in X are called closed sets. A subset of Xmay be neitherclosed nor open, either closed or open, or both. A set that is both closed and open is called a clopen set.

Examples

1. X = 1, 2, 3, 4 and collection τ = , 1, 2, 3, 4 of only the two subsets of X required by the axioms forma topology, the trivial topology (indiscrete topology).

2. X = 1, 2, 3, 4 and collection τ = , 2, 1, 2, 2, 3, 1, 2, 3, 1, 2, 3, 4 of six subsets of X formanother topology.

3. X = 1, 2, 3, 4 and collection τ = P(X) (the power set of X) form a third topology, the discrete topology.

4. X =Z, the set of integers, and collection τ equal to all finite subsets of the integers plusZ itself is not a topology,because (for example) the union of all finite sets not containing zero is infinite but is not all of Z, and so is notin τ .

74.2.3 Closed sets definition

Using de Morgan’s laws, the above axioms defining open sets become axioms defining closed sets:

1. The empty set and X are closed.

2. The intersection of any collection of closed sets is also closed.

3. The union of any finite number of closed sets is also closed.

Using these axioms, another way to define a topological space is as a set X together with a collection τ of closedsubsets of X. Thus the sets in the topology τ are the closed sets, and their complements in X are the open sets.

74.2.4 Other definitions

There are many other equivalent ways to define a topological space: in other words, the concepts of neighbourhoodor of open respectively closed set can be reconstructed from other starting points and satisfy the correct axioms.Another way to define a topological space is by using the Kuratowski closure axioms, which define the closed sets asthe fixed points of an operator on the power set of X.A net is a generalisation of the concept of sequence. A topology is completely determined if for every net in X theset of its accumulation points is specified.

74.3 Comparison of topologies

Main article: Comparison of topologies

A variety of topologies can be placed on a set to form a topological space. When every set in a topology τ1 is also ina topology τ2 and τ1 is a subset of τ2, we say that τ2 is finer than τ1, and τ1 is coarser than τ2. A proof that reliesonly on the existence of certain open sets will also hold for any finer topology, and similarly a proof that relies onlyon certain sets not being open applies to any coarser topology. The terms larger and smaller are sometimes used inplace of finer and coarser, respectively. The terms stronger and weaker are also used in the literature, but with littleagreement on the meaning, so one should always be sure of an author’s convention when reading.The collection of all topologies on a given fixed set X forms a complete lattice: if F = τα| α in A is a collectionof topologies on X, then the meet of F is the intersection of F, and the join of F is the meet of the collection of alltopologies on X that contain every member of F.

192 CHAPTER 74. TOPOLOGICAL SPACE

74.4 Continuous functions

A function f : X→ Y between topological spaces is called continuous if for all x ∈X and all neighbourhoodsN of f(x)there is a neighbourhoodM of x such that f(M) ⊆N. This relates easily to the usual definition in analysis. Equivalently,f is continuous if the inverse image of every open set is open.[5] This is an attempt to capture the intuition that thereare no “jumps” or “separations” in the function. A homeomorphism is a bijection that is continuous and whose inverseis also continuous. Two spaces are called homeomorphic if there exists a homeomorphism between them. From thestandpoint of topology, homeomorphic spaces are essentially identical.In category theory, Top, the category of topological spaces with topological spaces as objects and continuous functionsas morphisms is one of the fundamental categories in category theory. The attempt to classify the objects of thiscategory (up to homeomorphism) by invariants has motivated areas of research, such as homotopy theory, homologytheory, and K-theory etc.

74.5 Examples of topological spaces

A given set may have many different topologies. If a set is given a different topology, it is viewed as a differenttopological space. Any set can be given the discrete topology in which every subset is open. The only convergentsequences or nets in this topology are those that are eventually constant. Also, any set can be given the trivial topology(also called the indiscrete topology), in which only the empty set and the whole space are open. Every sequence andnet in this topology converges to every point of the space. This example shows that in general topological spaces,limits of sequences need not be unique. However, often topological spaces must be Hausdorff spaces where limitpoints are unique.There are many ways of defining a topology on R, the set of real numbers. The standard topology on R is generatedby the open intervals. The set of all open intervals forms a base or basis for the topology, meaning that every openset is a union of some collection of sets from the base. In particular, this means that a set is open if there exists anopen interval of non zero radius about every point in the set. More generally, the Euclidean spaces Rn can be givena topology. In the usual topology on Rn the basic open sets are the open balls. Similarly, C, the set of complexnumbers, and Cn have a standard topology in which the basic open sets are open balls.Every metric space can be given a metric topology, in which the basic open sets are open balls defined by the metric.This is the standard topology on any normed vector space. On a finite-dimensional vector space this topology is thesame for all norms.Many sets of linear operators in functional analysis are endowed with topologies that are defined by specifying whena particular sequence of functions converges to the zero function.Any local field has a topology native to it, and this can be extended to vector spaces over that field.Every manifold has a natural topology since it is locally Euclidean. Similarly, every simplex and every simplicialcomplex inherits a natural topology from Rn.The Zariski topology is defined algebraically on the spectrum of a ring or an algebraic variety. On Rn or Cn, theclosed sets of the Zariski topology are the solution sets of systems of polynomial equations.A linear graph has a natural topology that generalises many of the geometric aspects of graphs with vertices andedges.The Sierpiński space is the simplest non-discrete topological space. It has important relations to the theory of com-putation and semantics.There exist numerous topologies on any given finite set. Such spaces are called finite topological spaces. Finite spacesare sometimes used to provide examples or counterexamples to conjectures about topological spaces in general.Any set can be given the cofinite topology in which the open sets are the empty set and the sets whose complementis finite. This is the smallest T1 topology on any infinite set.Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or its complementis countable. When the set is uncountable, this topology serves as a counterexample in many situations.The real line can also be given the lower limit topology. Here, the basic open sets are the half open intervals [a, b).This topology on R is strictly finer than the Euclidean topology defined above; a sequence converges to a point in thistopology if and only if it converges from above in the Euclidean topology. This example shows that a set may have

74.6. TOPOLOGICAL CONSTRUCTIONS 193

many distinct topologies defined on it.If Γ is an ordinal number, then the set Γ = [0, Γ) may be endowed with the order topology generated by the intervals(a, b), [0, b) and (a, Γ) where a and b are elements of Γ.

74.6 Topological constructions

Every subset of a topological space can be given the subspace topology in which the open sets are the intersectionsof the open sets of the larger space with the subset. For any indexed family of topological spaces, the product can begiven the product topology, which is generated by the inverse images of open sets of the factors under the projectionmappings. For example, in finite products, a basis for the product topology consists of all products of open sets. Forinfinite products, there is the additional requirement that in a basic open set, all but finitely many of its projectionsare the entire space.A quotient space is defined as follows: if X is a topological space and Y is a set, and if f : X→ Y is a surjectivefunction, then the quotient topology on Y is the collection of subsets of Y that have open inverse images under f. Inother words, the quotient topology is the finest topology on Y for which f is continuous. A common example of aquotient topology is when an equivalence relation is defined on the topological space X. The map f is then the naturalprojection onto the set of equivalence classes.The Vietoris topology on the set of all non-empty subsets of a topological space X, named for Leopold Vietoris, isgenerated by the following basis: for every n-tuple U1, ..., Un of open sets in X, we construct a basis set consistingof all subsets of the union of the Ui that have non-empty intersections with each Ui.

74.7 Classification of topological spaces

Topological spaces can be broadly classified, up to homeomorphism, by their topological properties. A topologicalproperty is a property of spaces that is invariant under homeomorphisms. To prove that two spaces are not home-omorphic it is sufficient to find a topological property not shared by them. Examples of such properties includeconnectedness, compactness, and various separation axioms.See the article on topological properties for more details and examples.

74.8 Topological spaces with algebraic structure

For any algebraic objects we can introduce the discrete topology, under which the algebraic operations are continuousfunctions. For any such structure that is not finite, we often have a natural topology compatible with the algebraicoperations, in the sense that the algebraic operations are still continuous. This leads to concepts such as topologicalgroups, topological vector spaces, topological rings and local fields.

74.9 Topological spaces with order structure• Spectral. A space is spectral if and only if it is the prime spectrum of a ring (Hochster theorem).

• Specialization preorder. In a space the specialization (or canonical) preorder is defined by x ≤ y if andonly if clx ⊆ cly.

74.10 Specializations and generalizations

The following spaces and algebras are either more specialized or more general than the topological spaces discussedabove.

• Proximity spaces provide a notion of closeness of two sets.

194 CHAPTER 74. TOPOLOGICAL SPACE

• Metric spaces embody a metric, a precise notion of distance between points.

• Uniform spaces axiomatize ordering the distance between distinct points.

• A topological space in which the points are functions is called a function space.

• Cauchy spaces axiomatize the ability to test whether a net is Cauchy. Cauchy spaces provide a general settingfor studying completions.

• Convergence spaces capture some of the features of convergence of filters.

• Grothendieck sites are categories with additional data axiomatizing whether a family of arrows covers an object.Sites are a general setting for defining sheaves.

74.11 See also• Space (mathematics)

• Kolmogorov space (T0)

• accessible/Fréchet space (T1)

• Hausdorff space (T2)

• Completely Hausdorff space and Urysohn space (T₂½)

• Regular space and regular Hausdorff space (T3)

• Tychonoff space and completely regular space (T₃½)

• Normal Hausdorff space (T4)

• Completely normal Hausdorff space (T5)

• Perfectly normal Hausdorff space (T6)

• Quasitopological space

• Complete Heyting algebra – The system of all open sets of a given topological space ordered by inclusion is acomplete Heyting algebra.

74.12 Notes[1] Schubert 1968, p. 13

[2] Brown 2006, section 2.1.

[3] Brown 2006, section 2.2.

[4] Armstrong 1983, definition 2.1.

[5] Armstrong 1983, theorem 2.6.

74.13 References• Armstrong, M. A. (1983) [1979]. Basic Topology. Undergraduate Texts in Mathematics. Springer. ISBN0-387-90839-0.

• Bredon, Glen E., Topology and Geometry (Graduate Texts in Mathematics), Springer; 1st edition (October 17,1997). ISBN 0-387-97926-3.

• Bourbaki, Nicolas; Elements of Mathematics: General Topology, Addison-Wesley (1966).

74.14. EXTERNAL LINKS 195

• Brown, Ronald, Topology and groupoids, Booksurge (2006) ISBN 1-4196-2722-8 (3rd edition of differentlytitled books) (order from amazon.com).

• Čech, Eduard; Point Sets, Academic Press (1969).

• Fulton, William, Algebraic Topology, (Graduate Texts in Mathematics), Springer; 1st edition (September 5,1997). ISBN 0-387-94327-7.

• Lipschutz, Seymour; Schaum’s Outline of General Topology, McGraw-Hill; 1st edition (June 1, 1968). ISBN0-07-037988-2.

• Munkres, James; Topology, Prentice Hall; 2nd edition (December 28, 1999). ISBN 0-13-181629-2.

• Runde, Volker; A Taste of Topology (Universitext), Springer; 1st edition (July 6, 2005). ISBN 0-387-25790-X.

• Schubert, Horst (1968), Topology, Allyn and Bacon

• Steen, Lynn A. and Seebach, J. Arthur Jr.; Counterexamples in Topology, Holt, Rinehart and Winston (1970).ISBN 0-03-079485-4.

• Vaidyanathaswamy, R. (1960). Set Topology. Chelsea Publishing Co. ISBN 0486404560.

• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

74.14 External links• Hazewinkel, Michiel, ed. (2001), “Topological space”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Topological space at PlanetMath.org.

Chapter 75

Topologist’s sine curve

In the branch of mathematics known as topology, the topologist’s sine curve is a topological space with severalinteresting properties that make it an important textbook example.It can be defined as the graph of the function sin(1/x) on the half-open interval (0, 1], together with the origin, underthe topology induced from the Euclidean plane:

T =(

x, sin 1x

): x ∈ (0, 1]

∪ (0, 0).

75.1 Image of the curve

As x approaches zero from the right, the magnitude of the rate of change of 1/x increases. This is why the frequencyof the sine wave increases as one moves to the left in the graph.

75.2 Properties

The topologist’s sine curve T is connected but neither locally connected nor path connected. This is because it includesthe point (0,0) but there is no way to link the function to the origin so as to make a path.The space T is the continuous image of a locally compact space (namely, let V be the space −1 ∪ (0, 1], and usethe map f from V to T defined by f(−1) = (0,0) and f(x) = (x, sin(1/x)) for x > 0), but T is not locally compact itself.

196

75.3. VARIANTS 197

The topological dimension of T is 1.

75.3 Variants

Two variants of the topologist’s sine curve have other interesting properties.The closed topologist’s sine curve can be defined by taking the topologist’s sine curve and adding its set of limitpoints, (0, y) | y ∈ [−1, 1] . This space is closed and bounded and so compact by the Heine–Borel theorem, but hassimilar properties to the topologist’s sine curve—it too is connected but neither locally connected nor path-connected.The extended topologist’s sine curve can be defined by taking the closed topologist’s sine curve and adding to it theset (x, 1) | x ∈ [0, 1] . It is arc connected but not locally connected.

75.4 References• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of1978 ed.), Mineola, NY: Dover Publications, Inc., pp. 137–138, ISBN 978-0-486-68735-3, MR 1382863

• Weisstein, Eric W., “Topologist’s Sine Curve”, MathWorld.

Chapter 76

Trivial topology

In topology, a topological space with the trivial topology is one where the only open sets are the empty set and theentire space. Such a space is sometimes called an indiscrete space. Intuitively, this has the consequence that all pointsof the space are “lumped together” and cannot be distinguished by topological means; it belongs to a pseudometricspace in which the distance between any two points is zero.The trivial topology is the topology with the least possible number of open sets, since the definition of a topologyrequires these two sets to be open. Despite its simplicity, a space X with more than one element and the trivialtopology lacks a key desirable property: it is not a T0 space.Other properties of an indiscrete space X—many of which are quite unusual—include:

• The only closed sets are the empty set and X.

• The only possible basis of X is X.

• If X has more than one point, then since it is not T0, it does not satisfy any of the higher T axioms either. Inparticular, it is not a Hausdorff space. Not being Hausdorff, X is not an order topology, nor is it metrizable.

• X is, however, regular, completely regular, normal, and completely normal; all in a rather vacuous way though,since the only closed sets are ∅ and X.

• X is compact and therefore paracompact, Lindelöf, and locally compact.

• Every function whose domain is a topological space and codomain X is continuous.

• X is path-connected and so connected.

• X is second-countable, and therefore is first-countable, separable and Lindelöf.

• All subspaces of X have the trivial topology.

• All quotient spaces of X have the trivial topology

• Arbitrary products of trivial topological spaces, with either the product topology or box topology, have thetrivial topology.

• All sequences in X converge to every point of X. In particular, every sequence has a convergent subsequence(the whole sequence), thus X is sequentially compact.

• The interior of every set except X is empty.

• The closure of every non-empty subset of X is X. Put another way: every non-empty subset of X is dense, aproperty that characterizes trivial topological spaces.

• As a result of this, the closure of every open subset U of X is either ∅ (if U = ∅) or X (otherwise).In particular, the closure of every open subset of X is again an open set, and therefore X is extremallydisconnected.

198

76.1. SEE ALSO 199

• If S is any subset of X with more than one element, then all elements of X are limit points of S. If S is asingleton, then every point of X \ S is still a limit point of S.

• X is a Baire space.

• Two topological spaces carrying the trivial topology are homeomorphic iff they have the same cardinality.

In some sense the opposite of the trivial topology is the discrete topology, in which every subset is open.The trivial topology belongs to a uniform space in which the whole cartesian product X × X is the only entourage.Let Top be the category of topological spaces with continuous maps and Set be the category of sets with functions. IfF : Top→ Set is the functor that assigns to each topological space its underlying set (the so-called forgetful functor),and G : Set→ Top is the functor that puts the trivial topology on a given set, then G is right adjoint to F. (The functorH : Set→ Top that puts the discrete topology on a given set is left adjoint to F.)

76.1 See also• Triviality (mathematics)

76.2 References• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446

Chapter 77

Tychonoff plank

In topology, the Tychonoff plank is a topological space defined using ordinal spaces that is a counterexample toseveral plausible-sounding conjectures.

77.1 Definition

It is defined as the topological product of the two ordinal spaces [0, ω1] and [0, ω] , where ω is the first infinite ordinaland ω1 the first uncountable ordinal.

77.2 Deleted form

The deleted Tychonoff plank is obtained by deleting the point∞ = (ω1, ω) .

77.3 Properties

The Tychonoff plank is a compact Hausdorff space and is therefore a normal space. However, the deleted Tychonoffplank is non-normal. Therefore the Tychonoff plank is not completely normal. This shows that a subspace of a normalspace need not be normal. The Tychonoff plank is not perfectly normal because it is not a Gδ space: the singleton∞ is closed but not a Gδ set.

77.4 References• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446

77.5 External links• Barile, Margherita, “Tychonoff Plank”, MathWorld.

200

Chapter 78

Tychonoff space

In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds oftopological spaces. These conditions are examples of separation axioms.Tychonoff spaces are named after Andrey Nikolayevich Tychonoff, whose Russian name (Тихонов) is variouslyrendered as “Tychonov”, “Tikhonov”, “Tihonov”, “Tichonov” etc.

78.1 Definitions

Suppose that X is a topological space.X is a completely regular space if given any closed set F and any point x that does not belong to F, then there is acontinuous function f from X to the real line R such that f(x) is 0 and, for every y in F, f(y) is 1. In other terms, thiscondition says that x and F can be separated by a continuous function.X is a Tychonoff space, or T3½ space, or Tπ space, or completely T3 space if it is both completely regular andHausdorff.Note that some mathematical literature uses different definitions for the term “completely regular” and the termsinvolving “T”. The definitions that we have given here are the ones usually used today; however, some authors switchthe meanings of the two kinds of terms, or use all terms synonymously for only one condition. In Wikipedia, we willuse the terms “completely regular” and “Tychonoff” freely, but we'll avoid the less clear “T” terms. In other literature,you should take care to find out which definitions the author is using. (The phrase “completely regular Hausdorff”,however, is unambiguous, and always means a Tychonoff space.) For more on this issue, see History of the separationaxioms.Completely regular spaces and Tychonoff spaces are related through the notion of Kolmogorov equivalence. A topo-logical space is Tychonoff if and only if it’s both completely regular and T0. On the other hand, a space is completelyregular if and only if its Kolmogorov quotient is Tychonoff.

78.2 Examples and counterexamples

Almost every topological space studied in mathematical analysis is Tychonoff, or at least completely regular. Forexample, the real line is Tychonoff under the standard Euclidean topology. Other examples include:

• Every metric space is Tychonoff; every pseudometric space is completely regular.

• Every locally compact regular space is completely regular, and therefore every locally compact Hausdorff spaceis Tychonoff.

• In particular, every topological manifold is Tychonoff.

• Every totally ordered set with the order topology is Tychonoff.

• Every topological group is completely regular.

201

202 CHAPTER 78. TYCHONOFF SPACE

• Generalising both the metric spaces and the topological groups, every uniform space is completely regular. Theconverse is also true: every completely regular space is uniformisable.

• Every CW complex is Tychonoff.• Every normal regular space is completely regular, and every normal Hausdorff space is Tychonoff.• The Niemytzki plane is an example of a Tychonoff space which is not normal.

78.3 Properties

78.3.1 Preservation

Complete regularity and the Tychonoff property are well-behaved with respect to initial topologies. Specifically,complete regularity is preserved by taking arbitrary initial topologies and the Tychonoff property is preserved bytaking point-separating initial topologies. It follows that:

• Every subspace of a completely regular or Tychonoff space has the same property.• A nonempty product space is completely regular (resp. Tychonoff) if and only if each factor space is completelyregular (resp. Tychonoff).

Like all separation axioms, complete regularity is not preserved by taking final topologies. In particular, quotients ofcompletely regular spaces need not be regular. Quotients of Tychonoff spaces need not even be Hausdorff. Thereare closed quotients of the Moore plane which provide counterexamples.

78.3.2 Real-valued continuous functions

For any topological space X, let C(X) denote the family of real-valued continuous functions on X and let C*(X) bethe subset of bounded real-valued continuous functions.Completely regular spaces can be characterized by the fact that their topology is completely determined by C(X) orC*(X). In particular:

• A space X is completely regular if and only if it has the initial topology induced by C(X) or C*(X).• A space X is completely regular if and only if every closed set can be written as the intersection of a family ofzero sets in X (i.e. the zero sets form a basis for the closed sets of X).

• A space X is completely regular if and only if the cozero sets of X form a basis for the topology of X.

Given an arbitrary topological space (X, τ) there is a universal way of associating a completely regular space with (X,τ). Let ρ be the initial topology on X induced by Cτ(X) or, equivalently, the topology generated by the basis of cozerosets in (X, τ). Then ρ will be the finest completely regular topology on X which is coarser than τ. This constructionis universal in the sense that any continuous function

f : (X, τ) → Y

to a completely regular space Y will be continuous on (X, ρ). In the language of category theory, the functor whichsends (X, τ) to (X, ρ) is left adjoint to the inclusion functor CReg → Top. Thus the category of completely regularspaces CReg is a reflective subcategory of Top, the category of topological spaces. By taking Kolmogorov quotients,one sees that the subcategory of Tychonoff spaces is also reflective.One can show that Cτ(X) = Cᵨ(X) in the above construction so that the rings C(X) and C*(X) are typically onlystudied for completely regular spaces X.The category of real compact Tychonoff spaces is anti-equivalent to the category of the rings C(X) (where X is realcompact) together with ring homomorphisms as maps. For example one can reconstruct $X$ from C(X) when X is(real) compact. The algebraic theory of these rings is therefore subject of intensive studies. A vast generalisation ofthis class of rings which still resembles many properties of Tychonoff spaces but is also applicable in real algebraicgeometry, is the class of real closed rings.

78.4. REFERENCES 203

78.3.3 Embeddings

Tychonoff spaces are precisely those spaces which can be embedded in compact Hausdorff spaces. More precisely,for every Tychonoff space X, there exists a compact Hausdorff space K such that X is homeomorphic to a subspaceof K.In fact, one can always choose K to be a Tychonoff cube (i.e. a possibly infinite product of unit intervals). EveryTychonoff cube is compact Hausdorff as a consequence of Tychonoff’s theorem. Since every subspace of a compactHausdorff space is Tychonoff one has:

A topological space is Tychonoff if and only if it can be embedded in a Tychonoff cube.

78.3.4 Compactifications

Of particular interest are those embeddings where the image ofX is dense inK; these are calledHausdorff compactificationsof X. Given any embedding of a Tychonoff space X in a compact Hausdorff space K the closure of the image of Xin K is a compactification of X.Among those Hausdorff compactifications, there is a unique “most general” one, the Stone–Čech compactificationβX. It is characterised by the universal property that, given a continuous map f fromX to any other compact Hausdorffspace Y, there is a unique continuous map g from βX to Y that extends f in the sense that f is the composition of gand j.

78.3.5 Uniform structures

Complete regularity is exactly the condition necessary for the existence of uniform structures on a topological space.In other words, every uniform space has a completely regular topology and every completely regular space X isuniformizable. A topological space admits a separated uniform structure if and only if it is Tychonoff.Given a completely regular spaceX there is usually more than one uniformity onX that is compatible with the topologyof X. However, there will always be a finest compatible uniformity, called the fine uniformity on X. If X is Tychonoff,then the uniform structure can be chosen so that βX becomes the completion of the uniform space X.

78.4 References• Stephen Willard, General Topology, (1970) Addison-Wesley Publishing Company, Reading Massachusetts.

• Gillman, Leonard; Jerison, Meyer Rings of continuous functions. Reprint of the 1960 edition. Graduate Textsin Mathematics, No. 43. Springer-Verlag, New York-Heidelberg, 1976. xiii+300 pp

204 CHAPTER 78. TYCHONOFF SPACE

78.5 Text and image sources, contributors, and licenses

78.5.1 Text• A-paracompact space Source: https://en.wikipedia.org/wiki/A-paracompact_space?oldid=532052977 Contributors: Zundark, Paul Au-

gust, Vipul, Michael Slone, SmackBot, Silly rabbit, Lambiam, Cydebot, Ryan, JackSchmidt, Addbot, Brad7777 and Anonymous: 3• Adjunction space Source: https://en.wikipedia.org/wiki/Adjunction_space?oldid=613450359 Contributors: Zundark, TakuyaMurata,

Charles Matthews, Giftlite, Fropuff, Gauge, Oleg Alexandrov, Linas, BD2412, RonnieBrown, TheKMan, Tesseran, Keyi, Konradek,JustAGal, Subh83, JackSchmidt, Addbot, Brad7777 and Anonymous: 5

• Appert topology Source: https://en.wikipedia.org/wiki/Appert_topology?oldid=650693371Contributors: Addbot, Sławomir Biały, Masti-Bot, Fly by Night, Helpful Pixie Bot and Anonymous: 1

• Arens–Fort space Source: https://en.wikipedia.org/wiki/Arens%E2%80%93Fort_space?oldid=551323116Contributors: Zundark,MichaelHardy, Charles Matthews, Burn, Algebraist, SmackBot, Xanthoxyl, Ntsimp, Headbomb, LokiClock, JackSchmidt, Addbot, Yobot andLucienBOT

• Box topology Source: https://en.wikipedia.org/wiki/Box_topology?oldid=632990165Contributors: Zundark,Michael Hardy, JitseNiesen,Choni, Fropuff, Paul August, O18, SteinbDJ, Joriki, Linas, Algebraist, Melchoir, Bluebot, Dreadstar, Trumpet marietta 45750, Loki-Clock, Addbot, Topology Expert, SPat, Gebstadter, AnomieBOT, J04n, Point-set topologist, Tal physdancer, Tiled, ZéroBot, Faizan,999Bedbugs, Ereznesh and Anonymous: 9

• Cantor set Source: https://en.wikipedia.org/wiki/Cantor_set?oldid=686053516 Contributors: Damian Yerrick, AxelBoldt, Mav, Zun-dark, TheAnome, XJaM, Toby~enwiki, TobyBartels, Miguel~enwiki, Michael Hardy, Karada, Goatasaur, Schneelocke, CharlesMatthews,Dcoetzee, Nohat, Hyacinth, Pseudometric, AndrewKepert, Fibonacci, Marc Girod~enwiki, Robbot, Sverdrup, Choni, Bkell, Intangir,MOiRe, Aetheling, Ruakh, Dina, Tobias Bergemann, Tosha, Giftlite, Mshonle~enwiki, DavidCary, Fropuff, JeffBobFrank, Dmmaus,Mennucc, Fuzzy Logic, Sam Hocevar, Asbestos, Pyrop, TedPavlic, Gadykozma, Solkoll~enwiki, Marco Polo, Robotje, Townmouse,Eric Kvaalen, Complex01, Kotasik, Caesura, Derbeth, Lerdsuwa, Dzhim, Oleg Alexandrov, Hoziron, Simetrical, Linas, Plrk, Xaos-Bits, Cshirky, Mandarax, FlaBot, Mathbot, Xenobog, Fresheneesz, Chobot, YurikBot, RussBot, Zwobot, Tetracube, Vicarious, Jsnx,Selfworm, Neptunius, Chris the speller, Flyguy649, Kingdon, Daqu, Nishkid64, J. Finkelstein, Nicolas Bray, Loadmaster, Mets501,ILikeThings, CBM, Mattbuck, Reywas92, MC10, Robertinventor, Thijs!bot, Janviermichelle, Headbomb, Cj67, Salgueiro~enwiki, YKTimes, Beaumont, Magioladitis, Canter~enwiki, Cpiral, Policron, DavidCBryant, Adam1729, VolkovBot, TXiKiBoT, Cuinuc, Say some-thing then, Jesin, VanishedUserABC, Arcfrk, SieBot, Cwkmail, Swirlex, JackSchmidt, Blacklemon67, Readvanderbilt, Kompella, Ad-dbot, Mr.Xp, PV=nRT, ScAvenger, Soltanifar, Luckas-bot, Yobot, AnomieBOT, Materialscientist, DannyAsher, LilHelpa, Xqbot, Bdmy,RJGray, GrouchoBot, RibotBOT, LucienBOT,DrilBot, Sierpinksis, Pjmcswee, Dlivnat, EmausBot, Tagib, Ginger Conspiracy, Vikram360,Hlange1, EdoBot, F. Weckenbarth, Delusion23, Mesoderm, Helpful Pixie Bot, Mohsen.soltanifar, Banausbal, Geilamir, Envictis, Blevin-tron, BlevintronBot, Dexbot, Brirush, Tina.baghaee, Nigellwh, K9re11, Owenengine, Mohsen-Soltanifar-UofT, Muztre and Anonymous:99

• Cantor space Source: https://en.wikipedia.org/wiki/Cantor_space?oldid=674578263 Contributors: AxelBoldt, Michael Hardy, Timwi,Saltine, MathMartin, Giftlite, GraemeBartlett, MarkusKrötzsch, Fropuff, Terrible Tim, DemonThing, Tomgally, OlegAlexandrov, Linas,Jclemens, R.e.b., Hairy Dude, Trovatore, Kompik, SmackBot, Melchoir, Bluebot, PieRRoMaN, Infovoria, CenozoicEra, Iridescent, CBM,Epbr123, David Eppstein, Enviroboy, Tresiden, ClueBot, Yasmar, Cenarium, Luca Antonelli, Addbot, Jewelz 17, Yobot, AnomieBOT,Citation bot, HiW-Bot, D.Lazard, Solomon7968, Muztre and Anonymous: 19

• Comb space Source: https://en.wikipedia.org/wiki/Comb_space?oldid=685548007 Contributors: Michael Hardy, Tobias Bergemann,OdedSchramm, Salix alba, SmackBot, Adammajewski, Silly rabbit, David Eppstein, ImMAW, Paulburnett, JackSchmidt, Addbot, Topol-ogy Expert, Ironholds, Yobot, Citation bot, Tgoodwil, TjBot, Nosuchforever, Freeze S, Loxley and Anonymous: 7

• Compact convergence Source: https://en.wikipedia.org/wiki/Compact_convergence?oldid=544566130Contributors: MathMartin, Giftlite,Rich Farmbrough, Paul August, Momotaro, Simetrical, OdedSchramm, Scineram, RDBury, Od Mishehu, Silly rabbit, Jim.belk, Cydebot,Wikimorphism, Addbot, J04n, Erik9bot, Rickhev1, Brad7777 and Anonymous: 4

• Cosmic space Source: https://en.wikipedia.org/wiki/Cosmic_space?oldid=604283457 Contributors: Zundark, Michael Hardy, Paul Au-gust, Myasuda, Missvain, DH85868993, UnCatBot, Topology Expert, TheAMmollusc, Duffyt, Brad7777 and Deltahedron

• CW complex Source: https://en.wikipedia.org/wiki/CW_complex?oldid=674761063 Contributors: AxelBoldt, Michael Hardy, CharlesMatthews, Michael Larsen, Aenar, Altenmann, Tobias Bergemann, Giftlite, Graeme Bartlett, Lethe, Fropuff, Thufir Hawat, Gauge, Crust,Oleg Alexandrov, Salix alba, Gene.arboit, Michael Slone, Tong~enwiki, Ondenc, Marc Harper, AndrewWTaylor, Sardanaphalus, Smack-Bot, Bluebot, Nbarth, Vaughan Pratt, CRGreathouse, MotherFunctor, Turgidson, Robin S, LokiClock, Ambrose H. Field, MelchiorG,Rybu, YohanN7, Sun Creator, Pqnelson, Addbot, Fyrael, Cuaxdon, TravDogg, Luckas-bot, Yobot, Anne Bauval, FrescoBot, Lucien-BOT, D'ohBot, Lost-n-translation, Tgoodwil, WikitanvirBot, Felix Hoffmann, ZéroBot, ClueBot NG, TobiTobsensWiki, Nuermann,HUnTeR4subs, Brad7777, Cmalk, Mark viking, Vieque, Chinodennis, IjIzaR3KoSmYq and Anonymous: 38

• Discrete space Source: https://en.wikipedia.org/wiki/Discrete_space?oldid=686481200 Contributors: AxelBoldt, Toby Bartels, Patrick,Michael Hardy, Charles Matthews, Dcoetzee, Dysprosia, Jitse Niesen, Jose Ramos, MathMartin, Tobias Bergemann, Ancheta Wis, Tosha,BenFrantzDale, DefLog~enwiki, Baba, Paul August, Haham hanuka, Ricky81682, Oleg Alexandrov, Linas, Marudubshinki, Jshadias,Hairy Dude, Reyk, Arundhati bakshi, Curpsbot-unicodify, Ilmari Karonen, Poulpy, KnightRider~enwiki, Diegotorquemada, Mhss, Dread-star, Mwtoews, ArglebargleIV, Mets501, Jaybe~enwiki, JRSpriggs, CRGreathouse, Equendil, Cydebot, Thijs!bot, Salgueiro~enwiki, Alb-mont, VolkovBot, TXiKiBoT, Anonymous Dissident, AlleborgoBot, He7d3r, Marc van Leeuwen, Addbot, Topology Expert, Download,Luckas-bot, Yobot, Denispir, Ciphers, DannyAsher, FrescoBot, Cstanford.math, EmausBot, TuHan-Bot, Yukuairoy, Glmccolm, Qmv,Koertefa, Habitmelon, Brad7777, Deltahedron, TwoTwoHello and Anonymous: 26

• Discrete two-point space Source: https://en.wikipedia.org/wiki/Discrete_two-point_space?oldid=424981900Contributors: Michael Hardy,Graeme Bartlett and Anonymous: 1

• Dogbone space Source: https://en.wikipedia.org/wiki/Dogbone_space?oldid=647353154 Contributors: Michael Hardy, BD2412, R.e.b.,RDBury, Davidfsnyder, David Eppstein, LokiClock, CitationCleanerBot, ChrisGualtieri and Anonymous: 2

78.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 205

• Dunce hat (topology) Source: https://en.wikipedia.org/wiki/Dunce_hat_(topology)?oldid=661152541 Contributors: Michael Hardy,Fropuff, GalaxiaGuy, C S, BD2412, R.e.b., Gaius Cornelius, NeilN,Melchoir, Nbarth, Tawkerbot2, MotherFunctor, Turgidson, Rocchini,Steel1943, Yobot, Springsteen6, Hubert Shutrick, Jianluk91 and Anonymous: 4

• Equivariant topology Source: https://en.wikipedia.org/wiki/Equivariant_topology?oldid=663184218Contributors: TakuyaMurata, Framand Alvin Seville

• Erdős space Source: https://en.wikipedia.org/wiki/Erd%C5%91s_space?oldid=536743395 Contributors: Michael Hardy, Racklever,Ironholds, Zko28, BG19bot and Kephir

• Euclidean space Source: https://en.wikipedia.org/wiki/Euclidean_space?oldid=681868374 Contributors: AxelBoldt, Mav, Zundark,Tarquin, XJaM, Youandme, Tomo, Patrick, Michael Hardy, Dcljr, Karada, Looxix~enwiki, Angela, Charles Matthews, Dysprosia, Gren-delkhan, David Shay, MathMartin, Tobias Bergemann, Tosha, Giftlite, Lethe, Fropuff, Sriehl, DefLog~enwiki, Andycjp, Tomruen,Iantresman, Tzanko Matev, JohnArmagh, Rich Farmbrough, Paul August, Rgdboer, Msh210, Jimmycochrane, PAR, Eddie Dealtry,Dirac1933, Woohookitty, Isnow, Qwertyus, MarSch, MZMcBride, VKokielov, Kolbasz, Fresheneesz, NevilleDNZ, Chobot, Bgwhite,JPD, Wavelength, Hede2000, Epolk, KSmrq, SpuriousQ, ENeville, Mgnbar, Arthur Rubin, Brian Tvedt, RG2, JDspeeder1, SmackBot,Iamhove, Incnis Mrsi, Reedy, Mhss, JoeKearney, Silly rabbit, Hongooi, Tamfang, SashatoBot, Jim.belk, DabMachine, Dan Gluck, Kaarel,Yggdrasil014, Heqs, CmdrObot, GargoyleMT, Rudjek, Philomath3, Aiko, Guy Macon, Orionus, Salgueiro~enwiki, JAnDbot, Bencher-lite, CrizCraig, Magioladitis, TheChard, Avicennasis, Nucleophilic, Oderbolz, R'n'B, Reedy Bot, Policron, Trigamma, The enemies ofgod, Cerberus0, VolkovBot, IWhisky, Philip Trueman, Richardohio, WereSpielChequers, Da Joe, Caltas, Paolo.dL,MiNombreDeGuerra,Lightmouse, Denisarona, Tomas e, Mild Bill Hiccup, Gwguffey, Vsage, DhananSekhar, SilvonenBot, SkyLined, The Rationalist, Addbot,AkhtaBot, Pmod, Tide rolls, Legobot, Yobot, , Collieuk, Materialscientist, Citation bot, Sandip90, Xqbot, St.nerol, Nfr-Maat, Dead-clever23, RoyLeban, Ksuzanne, Mineralquelle, FrescoBot, Sławomir Biały, Alxeedo, RandomDSdevel, Gapato, Mikrosam Akademija2, Yunesj, Wikivictory, EmausBot, John of Reading, Quondum, Gbsrd, ClueBot NG, Wcherowi, Master Uegly, Cntras, Frank.manus,ElectricUvula, ElphiBot, MRG90, FeralOink, Userbot12, Lugia2453, Brirush, Limit-theorem, Eyesnore, Yardimsever, Tentinator, Fen-tonville, Mgkrupa, BemusedObserver, OrganicAltMetal, Ro4sho, Preethambittu, KasparBot, Dan6233 and Anonymous: 99

• Excluded point topology Source: https://en.wikipedia.org/wiki/Excluded_point_topology?oldid=505010277 Contributors: Toby Bar-tels, Charles Matthews, Tobias Bergemann, Fropuff, Jason Quinn, Paul August, Aquae, Bluebot, Jbolden1517, Headbomb, MetsBot,JackSchmidt, Lucinia~enwiki, HUnTeR4subs and Anonymous: 2

• Extension topology Source: https://en.wikipedia.org/wiki/Extension_topology?oldid=568515855 Contributors: Toby Bartels, Salix alba,Dmharvey, SmackBot, Tesseran, Jbolden1517, CBM,Headbomb, Raymondwinn, JackSchmidt, Lucinia~enwiki, EmausBot andMohamed-Ahmed-FG

• Finite topological space Source: https://en.wikipedia.org/wiki/Finite_topological_space?oldid=661353199 Contributors: Tobias Berge-mann, Fropuff, Jason Quinn, Bgwhite, SmackBot, CRGreathouse, Headbomb, Mathematrucker, Brusegadi, David Eppstein, Addbot,Luckas-bot, Yobot, Bethnim, Pawel8605, Helpful Pixie Bot, Mistory, Mogism, Howarth.kgs, Monkbot and Anonymous: 3

• First uncountable ordinal Source: https://en.wikipedia.org/wiki/First_uncountable_ordinal?oldid=663920033 Contributors: AxelBoldt,Michael Hardy, Aleph4, Tobias Bergemann, Touriste, RussBot, SmackBot, JRSpriggs, YohanN7, Addbot, Yobot, Kenilworth Terrace,Helpful Pixie Bot, Andyhowlett and Anonymous: 4

• Fixed-point space Source: https://en.wikipedia.org/wiki/Fixed-point_space?oldid=607955665Contributors: TheAnome,Michael Hardy,Silverfish, Charles Matthews, Oleg Alexandrov, Linas, Fnorp, Tony1, Cydebot, David Eppstein, Jesse V., EmausBot, Brad7777, JochenBurghardt, Mohamed-Ahmed-FG and Anonymous: 1

• Fort space Source: https://en.wikipedia.org/wiki/Fort_space?oldid=507627786 Contributors: Michael Hardy, Tobias Bergemann, DavidRadcliffe, Xanthoxyl, Ntsimp, Headbomb, JackSchmidt, WestwoodMatt, Sławomir Biały and Fly by Night

• Geometric topology (object) Source: https://en.wikipedia.org/wiki/Geometric_topology_(object)?oldid=635233230Contributors: CharlesMatthews, Zaslav, C S, Eubot, Trovatore, Bluebot, Delataur, David Eppstein, Jevansen, LarRan, Brirush and Mark viking

• Half-disk topology Source: https://en.wikipedia.org/wiki/Half-disk_topology?oldid=583655337 Contributors: Fly by Night, HelpfulPixie Bot and AmrithaJayadev1

• Hausdorff space Source: https://en.wikipedia.org/wiki/Hausdorff_space?oldid=647054341 Contributors: AxelBoldt, Magnus Manske,Zundark, Tarquin, Toby Bartels, B4hand, Michael Hardy, JakeVortex, TakuyaMurata, Ellywa, Cyp, BenKovitz, Ideyal, Charles Matthews,Dcoetzee, Dysprosia, Grendelkhan, Fibonacci, Robbot, MathMartin, Bkell, Tobias Bergemann, Tosha, Giftlite, Fropuff, Icairns, Vasile,Clarknova, TedPavlic, Guanabot, Westendgirl, Paul August, Bender235, El C, Vipul, .:Ajvol:., Tsirel, Dallashan~enwiki, Eric Kvaalen,Sligocki, Isaac, Jim Slim, Oleg Alexandrov, Linas, StradivariusTV, Graham87, BD2412, Chobot, YurikBot, Wavelength, Hairy Dude,Jessesaurus, Bota47, DVDRW, Sardanaphalus, Nbarth, JonAwbrey, Germandemat, Mets501, Dp462090, Cydebot, Alazaris, Dharma6662000,W3asal, Arcresu, RobHar, Salgueiro~enwiki, JAnDbot, AntiSpamBot, Policron, TXiKiBoT, Broadbot, Ocsenave, SieBot, OKBot, Ran-domblue, EconomicsGuy, Kruusamägi, DumZiBoT, Addbot, Mortense, Lightbot, Luckas-bot, Yobot, Ciphers, Citation bot, Kenneth-leebaker, Sławomir Biały, Vectornaut, Lapasotka, EmausBot, Drusus 0, Helpful Pixie Bot, Brad7777, MikeHaskel, Jochen Burghardt,Hierarchivist and Anonymous: 44

• Hawaiian earring Source: https://en.wikipedia.org/wiki/Hawaiian_earring?oldid=656626519 Contributors: Zundark, Michael Hardy,Nikai, Prumpf, Giftlite, Msh210, Alansohn, Mark.howison, Linas, RussBot, Gspr, Nbarth, Henning Makholm, Jim.belk, Gandalfxviv,Ranicki, Ntsimp, Oerjan, Top.Squark, Turgidson, Catgut, David Eppstein, Svick, Meowist, Citation bot 1, Electriccatfish2, Illia Connell,Debouch, Anrnusna and Anonymous: 17

• Hedgehog space Source: https://en.wikipedia.org/wiki/Hedgehog_space?oldid=609935504Contributors: Michael Hardy, CharlesMatthews,Tobias Bergemann, Linas, BD2412, Rjwilmsi, David Eppstein, LokiClock, Plclark, Addbot, Luckas-bot, EmausBot, Megrelm, HelpfulPixie Bot and Anonymous: 3

• Hilbert cube Source: https://en.wikipedia.org/wiki/Hilbert_cube?oldid=676858973 Contributors: Toby Bartels, Michael Hardy, Takuya-Murata, GTBacchus, Charles Matthews, MathMartin, Tobias Bergemann, Tosha, Giftlite, BenFrantzDale, Vivacissamamente, LindsayH,S.K., Iamunknown, Truthflux, Marudubshinki, Salix alba, Mathbot, Gadget850, Orthografer, Richard L. Peterson, Jheriko, WinBot,Javawizard, Marcosaedro, JackSchmidt, Alexey Muranov, Addbot, Anne Bauval, Point-set topologist, Citation bot 1, Trappist the monk,ZéroBot, Helpful Pixie Bot, Spaetzle, Solomon7968, Faizan and Anonymous: 18

206 CHAPTER 78. TYCHONOFF SPACE

• Hjalmar Ekdal topology Source: https://en.wikipedia.org/wiki/Hjalmar_Ekdal_topology?oldid=626901861 Contributors: Zundark,Rjwilmsi, SmackBot, Xanthoxyl, LokiClock, JackSchmidt, Eric-Wester, LilHelpa, Citation bot 1, PigFlu Oink, Trappist the monk andAnonymous: 1

• Homology sphere Source: https://en.wikipedia.org/wiki/Homology_sphere?oldid=677387941 Contributors: AxelBoldt, Patrick, Boud,Michael Hardy, Schneelocke, Charles Matthews, Rorro, Foxcub, Tosha, Giftlite, Fropuff, Mike40033, Gauge, C S, Msh210, OlegAlexandrov, Bkkbrad, Tabletop, BD2412, Rjwilmsi, R.e.b., Mathbot, Orthografer, JahJah, SmackBot, Originalbigj, Chris the speller,Tamfang, Jwy, Daqu, Akriasas, Jbergquist, Jim.belk, UncleDouggie, Dto, Headbomb, Hipster21, VectorPosse, Turgidson, David Epp-stein, SwordSmurf, Closenplay, Subh83, Marsupilamov, MystBot, Addbot, Nilesj, Debresser, Aldebaran66, MetaplecticGroup, Twri,Argumzio, Citation bot 1, Dinamik-bot, John of Reading, BattyBot, Saung Tadashi, Stomatapoll, Phleg1, Monkbot and Anonymous: 23

• Homotopy sphere Source: https://en.wikipedia.org/wiki/Homotopy_sphere?oldid=620683367 Contributors: Charles Matthews, Walt-pohl, C S, Msh210, Ceyockey, Bowman, Ryan Reich, Trovatore, CmdrObot, David Eppstein, Rybu, JerroldPease-Atlanta, Lightbot,Erik9bot, ChrisGualtieri and Anonymous: 6

• Hyperbolic space Source: https://en.wikipedia.org/wiki/Hyperbolic_space?oldid=686360579 Contributors: Zundark, Boud, Silverfish,Revolver, Daniel Quinlan, Giftlite, Gene Ward Smith, Fropuff, Eequor, Tomruen, Rgdboer, Eric Kvaalen, Alexrudd, Linas, BD2412,Mathbot, Woseph, SmackBot, Incnis Mrsi, Pokipsy76, Silly rabbit, Tamfang, Ulner, JRSpriggs, CRGreathouse, Dr.enh, Thijs!bot, No-clevername, David Eppstein, Sapphic, Arcfrk, Subh83, JerroldPease-Atlanta, Jtyard, Dthomsen8, Addbot, OBloodyHell, Yobot, TheEarwig, RibotBOT, Foobarnix, Calcyman, Sukarsono, Quondum, ClueBot NG, Justincheng12345-bot, 93, WillemienH and Anonymous:24

• Infinite broom Source: https://en.wikipedia.org/wiki/Infinite_broom?oldid=626902309 Contributors: Makyen, Xanthoxyl, RobHar, JJHarrison, David Eppstein, Derlay, LilHelpa, Citation bot 1, Trappist the monk, Helpful Pixie Bot and Anonymous: 1

• Infinite loop spacemachine Source: https://en.wikipedia.org/wiki/Infinite_loop_space_machine?oldid=592382349Contributors: Takuya-Murata, Bearcat, Magioladitis, AnomieBOT and Alvin Seville

• Interlocking interval topology Source: https://en.wikipedia.org/wiki/Interlocking_interval_topology?oldid=618447633 Contributors:Michael Hardy, Tobias Bergemann, Indeed123, Fly by Night, Helpful Pixie Bot, AK456 and Deltahedron

• Irrational winding of a torus Source: https://en.wikipedia.org/wiki/Irrational_winding_of_a_torus?oldid=621172629 Contributors:Michael Hardy, Dominus, Mandarax, BD2412, David Eppstein, Franp9am, Yobot, Kallikanzarid, Fly by Night and Helpful Pixie Bot

• K-topology Source: https://en.wikipedia.org/wiki/K-topology?oldid=674703203 Contributors: Michael Hardy, Paul August, BD2412,Dan131m, Chris the speller, Bwpach, Vanish2, LokiClock, JL-Bot, SchreiberBike, UnCatBot, Topology Expert, Asakura, BattyBot,ChrisGualtieri and Anonymous: 3

• Knaster–Kuratowski fan Source: https://en.wikipedia.org/wiki/Knaster%E2%80%93Kuratowski_fan?oldid=648052877 Contributors:Michael Hardy, Dfeuer, Tobias Bergemann, Tosha, Linas, R.e.b., Algebraist, Melchoir, Xanthoxyl, PamD, Sullivan.t.j, Smithers888,Austinmohr, Neworder1, JackSchmidt, UnCatBot, Addbot, LaaknorBot, Yobot, Xqbot, DrilBot, Trappist the monk, Uni.Liu and Anony-mous: 6

• Lexicographic order topology on the unit square Source: https://en.wikipedia.org/wiki/Lexicographic_order_topology_on_the_unit_square?oldid=626325738 Contributors: Michael Hardy, Tobias Bergemann, Mennucc, LokiClock, Iohannes Animosus, FrescoBot, Flyby Night, Helpful Pixie Bot and Anonymous: 4

• List of examples in general topology Source: https://en.wikipedia.org/wiki/List_of_examples_in_general_topology?oldid=674572418Contributors: Zundark, Dominus, Revolver, Charles Matthews, Aleph4, Fropuff, ZeroOne, Linas, R.e.b., Mathbot, Melchoir, Syrcatbot,Madmath789, Cydebot, Austinmohr, BabbaQ and K9re11

• Long line (topology) Source: https://en.wikipedia.org/wiki/Long_line_(topology)?oldid=678909395 Contributors: AxelBoldt, Zundark,Michael Hardy, Dominus, Saltine, Fibonacci, MathMartin, Tobias Bergemann, Fropuff, Gro-Tsen, Mennucc, Paul August, Nickj, Linas,Rjwilmsi, Mike Segal, R.e.b., HrHolm, YurikBot, Expensivehat, SmackBot, Melchoir, Michael Kinyon, Zero sharp, Sniffnoy, RogueN-inja, Magioladitis, David Eppstein, TXiKiBoT, Quilbert, Mike4ty4, JackSchmidt, Mr. Granger, Addbot, Roentgenium111, DOI bot,Jasper Deng, Luckas-bot, AnomieBOT, Citation bot 1, Trappist the monk, JordiGH, ClueBot NG, Wcherowi, Widr, Helpful Pixie Bot,Deltahedron, Lugia2453 and Anonymous: 21

• Loop space Source: https://en.wikipedia.org/wiki/Loop_space?oldid=607782988Contributors: CharlesMatthews, Lethe, Fropuff, Rgdboer,Oleg Alexandrov, Linas, MarSch, Staecker, R.e.b., Trovatore, JCSantos, Headbomb, VolkovBot, Katzmik, Subh83, Phe-bot, Desolate-Reality, Tilmanbauer, Addbot, Jonesey95, BG19bot, Brad7777, ChrisGualtieri, MrMorphism and Anonymous: 3

• Lower limit topology Source: https://en.wikipedia.org/wiki/Lower_limit_topology?oldid=626965941 Contributors: AxelBoldt, Zun-dark, Michael Hardy, Dominus, Loren Rosen, Charles Matthews, Dcoetzee, Dfeuer, Aleph4, Giftlite, Fropuff, TedPavlic, Paul August,Arthena, ABCD, Linas, Sullivan.t.j, JackSchmidt, Hans Adler, BOTarate, Addbot, Topology Expert, Luckas-bot, Yobot, 6cR, Trappistthe monk, Kephir, Mark viking, Geoffpointer and Anonymous: 7

• Menger sponge Source: https://en.wikipedia.org/wiki/Menger_sponge?oldid=683887935 Contributors: Bryan Derksen, XJaM, Heron,Michael Hardy, LittleDan, Glenn, Ghewgill, Schneelocke, Hyacinth, Sabbut, Phil Boswell, Stewartadcock, Rho~enwiki, Giftlite, Smjg,DavidCary, Fropuff, Everyking, Pyrop, Solkoll~enwiki, ACW, Arthena, Oleg Alexandrov, Linas, Waldir, Cshirky, Saperaud~enwiki,Mike Peel, Gareth McCaughan, R.e.b., John Baez, Mathbot, YurikBot, Gardar Rurak, Hellbus, Jessemerriman, Benandorsqueaks,Cmglee, Akumatatsu61, Bigbluefish, Eskimbot, Octahedron80, Emurphy42, A Geek Tragedy, Stepho-wrs, Henning Makholm, Wag-gers, Madmath789, Baserinia~enwiki, Mattbuck, A876, Michael C Price, Kylebinder, Thijs!bot, Janviermichelle, Davidhorman, Mar-garetWertheim, Santisan, Baccyak4H, Avicennasis, David Eppstein, Paiste42, CommonsDelinker, Patar knight, Prokofiev2, Supuhstar,Chiswick Chap, Lukax, D-Kuru, Taxiarchos228, Jameslwoodward, TXiKiBoT, Ironic Infidel, Spinningspark, Techman224, Pcmiller-wiki, Jwz, XLinkBot, Addbot, PV=nRT, Jarble, Mps, Luckas-bot, Yobot, 2D, AnomieBOT, Götz, JackieBot, Algorithme, Obersachse-bot, Misiekuk, Sławomir Biały, Steve Quinn, Kiefer.Wolfowitz, Karatelover1, Trappist the monk, Lotje, Ajraddatz, Slawekb, ZéroBot,999ers, Ontyx, ClueBot NG, Gareth Griffith-Jones, Soda drinker, David9550 and Anonymous: 69

• Metric space Source: https://en.wikipedia.org/wiki/Metric_space?oldid=685309437 Contributors: AxelBoldt, LC~enwiki, Zundark,Tarquin, XJaM, Toby Bartels, Edemaine, Paul Ebermann, Tomo, Patrick, Chas zzz brown, Michael Hardy, Wshun, SGBailey, Takuya-Murata, Looxix~enwiki, Andres, Tristanb, Ideyal, Revolver, RodC, CharlesMatthews, Dcoetzee, Dfeuer, Dysprosia, Jitse Niesen, Prumpf,Saltine, AndrewKepert, Mtcv, AnanthaRaman, Donarreiskoffer, Robbot, RedWolf, Altenmann, Romanm,MathMartin, Robinh, Aetheling,

78.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 207

Tobias Bergemann, Tosha, Giftlite, Bob Palin, Gene Ward Smith, Markus Krötzsch, Lupin, Fropuff, David Johnson, Python eggs, Gub-bubu, LiDaobing, Vivacissamamente, GamerRemag12@aol.com, Pyrop, TedPavlic, Guanabot, Paul August, SpookyMulder, BACbKA,Kinitawowi, El C, Rgdboer, Crisófilax, Miraage, Blotwell, Emhoo~enwiki, Obradovic Goran, Helix84, Tsirel, Jumbuck, Eric Kvaalen,Sligocki, Fiedorow, Themillofkeytone, Pashi, Kbolino, Oleg Alexandrov, Saeed, Joriki, Linas, Tabletop, Plowboylifestyle, Nileshbansal,Marudubshinki, Graham87, BD2412, Salix alba, Brighterorange, Mathbot, Margosbot~enwiki, Vulturejoe, Jenny Harrison, CiaPan,Chobot, YurikBot, Hairy Dude, Calumny, Number 57, Stefan Udrea, Kompik, Nothlit, JahJah, Pred, MullerHolk, RonnieBrown, Sar-danaphalus, SmackBot, Incnis Mrsi, Hammerite, PJTraill, Complexica, Nbarth, Hve, Xyzzy n, Meni Rosenfeld, Hyperwired, Mets501,Madmath789, Buckyboy314, Roland.barrat, CRGreathouse, KerryVeenstra, CmdrObot, Jackzhp, Thijs!bot, Epbr123, Kilva, Rlupsa,Seanskye, Urdutext, Orionus, Azaghal of Belegost, Dougher, JAnDbot, MER-C, Quentar~enwiki, Douglas Whitaker, Extropian314,Magioladitis, Wlod, JJ Harrison, David Eppstein, Oravec, MartinBot, TheSeven, Policron, Undernearththeman, GSpeight, LokiClock,Moswiki, TXiKiBoT, Plclark, Wikimorphism, Chirpstation, CenturionZ 1, Psymun747, SieBot, YonaBot, WereSpielChequers, Garde,Paolo.dL, MiNombreDeGuerra, Jorgen W, Skeptical scientist, Anchor Link Bot, Melcombe, Cliff, UKoch, Lbertolotti, Hans Adler,Vanished user tj23rpoij4tikkd, DumZiBoT, XLinkBot, Charles Sturm, Addbot, Tjlaxs, Haruth, LaaknorBot, Dyaa, Okcash, Zorrobot,Luckas-bot, Yobot, Ciphers, Joule36e5, Bdmy, Syena, Defeng.wu, DrilBot, HRoestBot, Kiefer.Wolfowitz, Sra2114, SpaceFlight89, Lotje,Jesse V., Biker333, Slawekb, Bethnim, Josve05a, Chharvey, SporkBot, L Kensington, ResearchRave, Reineke80, Wcherowi, Kstouras,Lifeonahilltop, Vinícius Machado Vogt, Helpful Pixie Bot, Tasky2, AdventurousSquirrel, Brad7777, Darvii, Lolmid, Teddyktchan, Ver-dana Bold, GeoffreyT2000, Harryalerta, KasparBot and Anonymous: 105

• Moore plane Source: https://en.wikipedia.org/wiki/Moore_plane?oldid=627019808 Contributors: Aleph4, Giftlite, Waltpohl, Yuvalmadar, Linas, Rjwilmsi, Algebraist, YurikBot, Tajmahall, SmackBot, Madmath789, CBM, Thijs!bot, Alphachimpbot, Marcosaedro,JackSchmidt, Mild Bill Hiccup, Addbot, Topology Expert, Yobot, Trappist the monk, Agalets and Anonymous: 5

• Nilpotent space Source: https://en.wikipedia.org/wiki/Nilpotent_space?oldid=598976452 Contributors: TakuyaMurata and David Epp-stein

• Overlapping interval topology Source: https://en.wikipedia.org/wiki/Overlapping_interval_topology?oldid=627024806 Contributors:Patrick, Tobias Bergemann, Linas, JackSchmidt and Trappist the monk

• Partially ordered space Source: https://en.wikipedia.org/wiki/Partially_ordered_space?oldid=663538398 Contributors: Michael Hardy,Bearcat, Eastlaw, A3nm, David Eppstein, AnomieBOT, Jakito, Jesse V., Paolo Lipparini and Mark Arsten

• Particular point topology Source: https://en.wikipedia.org/wiki/Particular_point_topology?oldid=627024896Contributors: Patrick, Ci-phergoth, Fibonacci, Tobias Bergemann, Fropuff, Ben Standeven, Linas, Ilmari Karonen, Tesseran, Physis, Mets501, Shenron, Jbolden1517,JackSchmidt, Topology Expert, Yobot, Trappist the monk, HUnTeR4subs and Anonymous: 3

• Partition topology Source: https://en.wikipedia.org/wiki/Partition_topology?oldid=627024921 Contributors: Michael Hardy, TobiasBergemann, Fropuff, TheObtuseAngleOfDoom, Sango123, Jbolden1517, MartinBot, UnrulyMoose, JackSchmidt, Trappist the monk,JewMad, Mark viking and Anonymous: 2

• Pointed space Source: https://en.wikipedia.org/wiki/Pointed_space?oldid=618910249 Contributors: Paul A, Charles Matthews, TobiasBergemann, Fropuff, Rich Farmbrough, Oleg Alexandrov, Linas, BD2412, RonnieBrown, SmackBot, RDBury, Addbot, Numbo3-bot,AnomieBOT, Erik9bot, ChuispastonBot, Omarct1989, Brad7777, Monkbot, JMP EAX and Anonymous: 6

• Pointwise convergence Source: https://en.wikipedia.org/wiki/Pointwise_convergence?oldid=680689270 Contributors: Michael Hardy,Gandalf61, Aetheling, Lethe, Paul August, Oleg Alexandrov, Joriki, Linas, FlaBot, YurikBot, Bullzeye, SmackBot, Od Mishehu, Eskim-bot, Phamkimson, Vina-iwbot~enwiki, Jim.belk, Stotr~enwiki, Cydebot, Albmont, Sullivan.t.j, Slimeknight, Wikimorphism, Blake3522,Thehotelambush, Maimai009, Addbot, Luckas-bot, Erel Segal, Rubinbot, Xqbot, Erik9bot, BenzolBot, DrilBot, MastiBot, IXhdBAH,Brad7777, YFdyh-bot, JYBot, Limit-theorem and Anonymous: 26

• Priestley space Source: https://en.wikipedia.org/wiki/Priestley_space?oldid=542641089Contributors: Michael Hardy, RobHar, Jmath666,Bearsona, Yobot, Gabelaia and Brad7777

• Projectively extended real line Source: https://en.wikipedia.org/wiki/Projectively_extended_real_line?oldid=674402096 Contributors:DamianYerrick, AxelBoldt, Zundark, Patrick,Michael Hardy, TakuyaMurata, CharlesMatthews, Giftlite, Fropuff, DemonThing, Rgdboer,Oleg Alexandrov, Jeff02, Luk, Patrickneil, Meni Rosenfeld, Mets501, Asyndeton, Vaughan Pratt, CRGreathouse, CBM, He Who Is,Knotwork, TechnoFaye, J.delanoy, Steven J. Anderson, Alpha Beta Epsilon, Addbot, Yobot, AnomieBOT, Tkuvho, Thinking of England,Quondum, D.Lazard, SporkBot, Llightex, IdealOmnicience, Wcherowi, Robsongr, SmartPerson123 and Anonymous: 17

• Prüfermanifold Source: https://en.wikipedia.org/wiki/Pr%C3%BCfer_manifold?oldid=648783253Contributors: Michael Hardy, R.e.b.,ShelfSkewed, Headbomb, Yobot, Sławomir Biały and K9re11

• Pseudocircle Source: https://en.wikipedia.org/wiki/Pseudocircle?oldid=664804202 Contributors: Toby Bartels, Charles Matthews, To-bias Bergemann, Fropuff, Fiedorow, Bgwhite, Kompik, RonnieBrown, Adavidb, JoeSperrazza, HUnTeR4subs and Anonymous: 1

• Pseudomanifold Source: https://en.wikipedia.org/wiki/Pseudomanifold?oldid=673641432Contributors: Michael Hardy, Bearcat, Tosha,Xezbeth, BD2412, Rjwilmsi, JohnBlackburne, Yobot, Fly by Night, Helpful Pixie Bot, Jianluk91, Equinox and Anonymous: 1

• Ran space Source: https://en.wikipedia.org/wiki/Ran_space?oldid=607705670 Contributors: Michael Hardy, TakuyaMurata, Bearcat,Jllm06, R'n'B and Deltahedron

• Real line Source: https://en.wikipedia.org/wiki/Real_line?oldid=682153923 Contributors: AxelBoldt, Toby Bartels, Michael Hardy, An-dres, Charles Matthews, Dcoetzee, Fibonacci, MathMartin, Tobias Bergemann, Giftlite, Ævar Arnfjörð Bjarmason, Jason Quinn, Kle-men Kocjancic, Rgdboer, Jumbuck, Oleg Alexandrov, Isnow, Marudubshinki, Jshadias, YurikBot, Kronocide, SmackBot, Silly rabbit,Jim.belk, CRGreathouse, Thijs!bot, Konradek, Magioladitis, Jmajeremy, Billinghurst, Paolo.dL, Estirabot, Addbot, Mrchapel0203, Laa-knorBot, Ciphers, Isheden, Omnipaedista, Erik9bot, Tkuvho, John of Reading, D.Lazard, Txus.aparicio, MelbourneStar, Stephan Kulla,Yardimsever, Dimension10 and Anonymous: 12

• Rose (topology) Source: https://en.wikipedia.org/wiki/Rose_(topology)?oldid=613452610Contributors: Zundark, Fropuff, Gauge, Linas,BD2412, Algebraist, Trovatore, Dreadstar, Akriasas, Jim.belk, Turgidson, David Eppstein, MystBot, Addbot, Compsonheir, ZéroBot, Ci-tationCleanerBot and Anonymous: 3

• Shrinking space Source: https://en.wikipedia.org/wiki/Shrinking_space?oldid=648199447 Contributors: Vivacissamamente, D6, Vipul,Cydebot, Vanish2, David Eppstein, Brad7777 and Noix07

208 CHAPTER 78. TYCHONOFF SPACE

• Sierpinski carpet Source: https://en.wikipedia.org/wiki/Sierpinski_carpet?oldid=661235147 Contributors: Damian Yerrick, Brion VIB-BER, Eloquence, Bryan Derksen, Piotr Gasiorowski, Michael Hardy, Kragen, Jengod, Charles Matthews, Hyacinth, Altenmann, Jor,Cyrius, Tosha, Giftlite, Herbee, Beland, Sam Hocevar, Rich Farmbrough, Gadykozma, Solkoll~enwiki, ABCD, Comrade009, Kgashok,Uffish, Oleg Alexandrov, Linas, GregorB, Cshirky, Qwertyus, Saperaud~enwiki, Mikedelsol, R.e.b., John Baez, Chobot, Seth Terashima,Roboto de Ajvol, YurikBot, Nol Aders, Emuka~enwiki, Ejdzej, CLW, Open2universe, SmackBot, F, Eskimbot, Ohnoitsjamie, Thumper-ward, PrimeHunter, Tamfang, OrphanBot, Henning Makholm, Gruznov, CRGreathouse, Thijs!bot, D4g0thur, Al Lemos, Escarbot,Barek, Beaumont, Sushant gupta, Lukeaw, David Eppstein, Prokofiev2, TXiKiBoT, Themcman1, Jamelan, Dmcq, Mwn3d, Jlrodri, Snig-brook, Joshi1983, Pernambuko, PixelBot, MystBot, Addbot, Luckas-bot, Erel Segal, Xqbot, LitRidl, LucienBOT,Getspaper, 00Ragora00,Mspreij, EmausBot, Didym, Romandarosa10, BG19bot, Northamerica1000, Professorjohnas, Deltahedron, Brirush, KarocksOrkav, Pi-otrGrochowski000 and Anonymous: 45

• Sierpinski triangle Source: https://en.wikipedia.org/wiki/Sierpinski_triangle?oldid=686066628 Contributors: Damian Yerrick, Axel-Boldt, Brion VIBBER, Eloquence, Mav, Bryan Derksen, Tarquin, Josh Grosse, Nonenmac, Michael Hardy, SGBailey, Cyp, Kragen,Evercat, Schneelocke, Dino, Jitse Niesen, Furrykef, Hyacinth, .mau., Carbuncle, Sander123, Altenmann, Wereon, SoLando, TobiasBergemann, Tosha, Giftlite, Herbee, Fropuff, Wikibob, PrisonerOfPain, Matt Crypto, Chowbok, Beland, Daniel,levine, Peter bertok,Kate, Discospinster, NeuronExMachina, Solkoll~enwiki, ESkog, El C, Bobo192, Marco Polo, .:Ajvol:., Russ3Z, TheParanoidOne, Oliv-erlewis, Derumi, Sligocki, Burn, Bart133, Yipdw, Pontus, RogerBarnett, Bsadowski1, DV8 2XL, Agutie, Oleg Alexandrov, Linas, Shree-vatsa, PoccilScript, Jfr26, A3r0, Saperaud~enwiki, 1wheel, Mathbot, Chobot, Volunteer Marek, YurikBot, Wavelength, PiAndWhipped-Cream, Nol Aders, DanielKO, Gaius Cornelius, Emuka~enwiki, NawlinWiki, Rick Norwood, Goffrie, Slicing, KGasso, Kungfuadam,Robertd, SmackBot, RDBury, Kslays, Eloil, Algumacoisaqq~enwiki, PrimeHunter, Mohan1986, Modest Genius, Simpsons contributor,Newmanbe, JonHarder, Sjb0926, Charles Merriam, Mexinadian, Pgc512, Acdx, Ozhiker, Mystìc, Mgiganteus1, WhiteShark, Rainwar-rior, Condem, Yodin, Tó campos~enwiki, Aquarin, DavidOaks, Phoenixrod, Mwoollams, Anthony Bradbury, Rev. Jack Zall, Dynaflow,Crandom, Thijs!bot, Epbr123, Sobreira, Marek69, I do not exist, Escarbot, Mhaitham.shammaa, Pikalax, Pichote, Dman727, JAnDbot,Beaumont, Lukeaw, David Eppstein, DerHexer, Gwern, MartinBot, Gcranston, CommonsDelinker, Ampersand17, J.delanoy, Prokofiev2,Maurice Carbonaro, Supuhstar, Policron, 28bytes, Fnlfntsyfn, Zoharby, Philip Trueman, TXiKiBoT, PaulTanenbaum, PDFbot, Mouseis back, MartinPackerIBM, Rajeevmenon, Alexkrules, Spinningspark, Dmcq, Joke758, SheepNotGoats, Tarun.Varshney, Eouw0o83hf,Svick, Anchor Link Bot, Denisarona, Martarius, ClueBot, Mym-uk, Justin W Smith, Arakunem, Joshi1983, Doseiai2, Jwz, Pernambuko,Phileasson, Excirial, Watchduck, PixelBot, Versus22, SoxBot III, LordBreetai, XLinkBot, Rror, FranciscoPadillaGarcia, Sickler77, Ad-dbot, Cooldudes187, MrOllie, Download, AgadaUrbanit, Lightbot, Luckas-bot, Yobot, Mintrick, The Ridger, Ivan Kuckir, Xqbot, Van-ryn21, TrulyEpic, Shadowjams, FrescoBot, Pepper, OgreBot, AstaBOTh15, Pekayer11, HRoestBot, WikiAntPedia, Backyardbronze,Syockit, PiRSquared17, Julio144, Avikar1, Vrenator, Muladhara, Sierpinksis, Phlegat, EmausBot, John of Reading, GoingBatty, Lateg,Ontyx, ClueBot NG, Aus.lavigne, Frietjes, KirbyRider, Ariel C.M.K., BG19bot, MusikAnimal, Dms00201, Akarpe, AndrewProgram-mer, Phigknotpig, Dexbot, Deltahedron, Brirush, The editor987, ZestyTaco386, Samuel Alex Potter, Mingopher, Sshockley44, Sarr Cat,KasparBot, SambaTriplets, Sappl, Swmmng and Anonymous: 182

• Sierpiński space Source: https://en.wikipedia.org/wiki/Sierpi%C5%84ski_space?oldid=678123874 Contributors: Zundark, MichaelHardy, Dominus, Revolver, Astronautics~enwiki, Tosha, Fropuff, Sade, Linas, R.e.b., Mathbot, Hairy Dude, Lenthe, Kompik, Crys-tallina, Incnis Mrsi, Jbolden1517, RogierBrussee, Plclark, JackSchmidt, Addbot, Yobot, KamikazeBot, ArthurBot, Xqbot, MastiBot,Trappist the monk, EmausBot, ZéroBot, Irancplusplus, JAaron95 and Anonymous: 4

• Simplicial complex Source: https://en.wikipedia.org/wiki/Simplicial_complex?oldid=684731537Contributors: Tomo, CharlesMatthews,1984, Altenmann, Ashwin, Tosha, Giftlite, BenFrantzDale, Tomruen, TedPavlic, Zaslav, Gauge, Oleg Alexandrov, Staecker, FlaBot,YurikBot, Gene.arboit, Gadget850, Sardanaphalus, SmackBot, UU, Henning Makholm, Mathsci, Equendil, Cydebot, Michael Fourman,Thijs!bot, D Haggerty, .anacondabot, Magioladitis, David Eppstein, Smithers888, VolkovBot, Trevorgoodchild, Neparis, PixelBot, Ad-dbot, Luckas-bot, Cflm001, Af2125, Erel Segal, Drilnoth, Control.valve, Prijutme4ty, Undsoweiter, Molitorppd22, Citation bot 1, Jowafan, EmausBot, Zzzgggrrr, Frietjes, BTotaro, BG19bot, Brad7777, Samreid94, KasparBot and Anonymous: 24

• Simplicial space Source: https://en.wikipedia.org/wiki/Simplicial_space?oldid=646556641 Contributors: TakuyaMurata, David Epp-stein, AnomieBOT, Cavarrone, Alvin Seville, ChrisGualtieri and Hijigne

• Smith–Volterra–Cantor set Source: https://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%93Cantor_set?oldid=684863115Contributors: Michael Hardy, Charles Matthews, Henrygb, Tobias Bergemann, Tosha, Lethe, Squash, Gauge, Tsirel, Linas, Henry Bottom-ley, Fresheneesz, Tetracube, AndrewWTaylor, Daqu, Mr Death, Mets501, Colin Rowat, David Eppstein, Ernest lk lam, Addbot, Uncia,Yobot, ArthurBot, Xqbot, GrouchoBot, LucienBOT, Thomassteinke and Anonymous: 12

• Sorgenfrey plane Source: https://en.wikipedia.org/wiki/Sorgenfrey_plane?oldid=627082456 Contributors: Michael Hardy, Dominus,MathMartin, Giftlite, Fropuff, Trevor MacInnis, Hairy Dude, Hennobrandsma, Od Mishehu, CBM, CharlotteWebb, Darklilac, Vanish2,Sullivan.t.j, Dimpas, JackSchmidt, Addbot, Topology Expert, Luckas-bot, LucienBOT, Citation bot 1, Trappist the monk, Alph Bot andAnonymous: 4

• Split interval Source: https://en.wikipedia.org/wiki/Split_interval?oldid=571710020 Contributors: Melchoir, David Eppstein, Markviking and Anonymous: 1

• Topologicalmonoid Source: https://en.wikipedia.org/wiki/Topological_monoid?oldid=674292577Contributors: TakuyaMurata, K9re11and KindleinKrüger

• Topological space Source: https://en.wikipedia.org/wiki/Topological_space?oldid=685445008Contributors: AxelBoldt, Zundark, XJaM,Toby Bartels, Olivier, Patrick, Michael Hardy, Wshun, Kku, Dineshjk, Karada, Hashar, Zhaoway~enwiki, Revolver, Charles Matthews,Dcoetzee, Dysprosia, Kbk, Taxman, Phys, Robbot, Nizmogtr, Fredrik, Saaska, MathMartin, P0lyglut, Tobias Bergemann, Giftlite, GeneWard Smith, Lethe, Fropuff, Dratman, DefLog~enwiki, Rhobite, Luqui, Paul August, Dolda2000, Elwikipedista~enwiki, Tompw, Aude,SgtThroat, Tsirel, Marc van Woerkom, Varuna, Kuratowski’s Ghost, Msh210, Keenan Pepper, Danog, Sligocki, Spambit, Oleg Alexan-drov, Woohookitty, Graham87, BD2412, Grammarbot, FlaBot, Sunayana, Tillmo, Chobot, Algebraist, YurikBot, Wavelength, HairyDude, NawlinWiki, Rick Norwood, Bota47, Stefan Udrea, Hirak 99, Arthur Rubin, Lendu, JoanneB, Eaefremov, RonnieBrown, Sar-danaphalus, SmackBot, Maksim-e~enwiki, Sciyoshi~enwiki, DHN-bot~enwiki, Tsca.bot, Tschwenn, LkNsngth, Vriullop, Arialblack, Iri-descent, Devourer09,Mattbr, AndrewDelong, Kupirijo, Roccorossi, Xantharius, Thijs!bot, Konradek, Odoncaoa, Escarbot, Salgueiro~enwiki,JAnDbot, YK Times, Bkpsusmitaa, Jakob.scholbach, Bbi5291, Wdevauld, J.delanoy, Pharaoh of the Wizards, Maurice Carbonaro, TheMudge, Jmajeremy, Policron, TXiKiBoT, Anonymous Dissident, Plclark, Aaron Rotenberg, Jesin, Arcfrk, SieBot, MiNombreDeGuerra,JerroldPease-Atlanta, JackSchmidt, Failure.exe, Egmontaz, Palnot, SilvonenBot, Addbot, CarsracBot, AnnaFrance, ChenzwBot, Luckas-bot, Yobot, SwisterTwister, AnomieBOT, Ciphers, Materialscientist, Citation bot, DannyAsher, FlordiaSunshine342, J04n, Point-set

78.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 209

topologist, FrescoBot, Jschnur, Jeroen De Dauw, TobeBot, Seahorseruler, Skakkle, Cstanford.math, ZéroBot, Chharvey, Wikfr, OrangeSuede Sofa, Liuthar, ClueBot NG, Wcherowi, Mesoderm, Vinícius Machado Vogt, Helpful Pixie Bot, Gaurav Nirala, Tom.hirschowitz,Pacerier, Solomon7968, Cpatra1984, Brad7777, Minsbot, LoganFromSA, MikeHaskel, Acer4666, Freeze S, Mark viking, Epicgenius,Kurt Artindagi, Improbable keeler, Amonk1962, GeoffreyT2000, KasparBot and Anonymous: 111

• Topologist’s sine curve Source: https://en.wikipedia.org/wiki/Topologist’s_sine_curve?oldid=685515367 Contributors: Edemaine,Michael Hardy, BenKovitz, Loren Rosen, Dcoetzee, Dysprosia, Sabbut, Morn, Josh Cherry, MathMartin, Tobias Bergemann, Marcika,Fropuff, Frencheigh, LucasVB, NickJohnson, Yuval madar, Paul August, Danog, Linas, HannsEwald, Nowhither, Saric, Adam majewski,Jim.belk, Ntsimp, RobHar, VoABot II, Pomte, JackSchmidt, Jane Bennet, Albambot, Addbot, Topology Expert, Luckas-bot, Raulshc,Ebony Jackson, RedBot, Trappist the monk, ZéroBot, Ørsted and Anonymous: 7

• Trivial topology Source: https://en.wikipedia.org/wiki/Trivial_topology?oldid=651331293 Contributors: AxelBoldt, Zundark, Patrick,Michael Hardy, Dcoetzee, Dysprosia, Jitse Niesen, David Shay, Lumos3, Fredrik, MathMartin, Tobias Bergemann, Tosha, Fropuff, Walt-pohl, Jason Quinn, Paul August, ABCD, Linas, BD2412, Jshadias, Nowhither, YurikBot, Poulpy, Mhss, Thijs!bot, Proximo.xv~enwiki,Salgueiro~enwiki, Trumpet marietta 45750, JackSchmidt, Ali Esfandiari, MystBot, Addbot, Luckas-bot, Yobot, KamikazeBot, Rubinbot,Trappist the monk and Anonymous: 8

• Tychonoff plank Source: https://en.wikipedia.org/wiki/Tychonoff_plank?oldid=638205660 Contributors: AxelBoldt, Dominus, TobiasBergemann, Giftlite, Fropuff, Waltpohl, Oleg Alexandrov, Gandalfxviv, CRGreathouse, Vanish2, David Eppstein, JackSchmidt, Addbot,Trappist the monk, Brad7777, Justincheng12345-bot, Brirush and Anonymous: 2

• Tychonoff space Source: https://en.wikipedia.org/wiki/Tychonoff_space?oldid=632689033 Contributors: AxelBoldt, Zundark, TobyBartels, Zhaoway~enwiki, Dysprosia, Fibonacci, Tosha, Abiola Lapite, Fropuff, Paul August, Linas, BD2412, Jshadias, Algebraist, DVDR W, Sardanaphalus, Mhss, Bluebot, Germandemat, Mets501, Stotr~enwiki, Equendil, NewEnglandYankee, Sigmundur, Agricola44,SieBot, Stca74, PipepBot, Luca Antonelli, Addbot, Мыша, Luckas-bot, Yobot, LucienBOT, EmausBot, Marcus0107, ZéroBot, PaoloLipparini, Brad7777 and Anonymous: 12

78.5.2 Images• File:45,_-315,_and_405_co-terminal_angles.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/56/45%2C_-315%2C_

and_405_co-terminal_angles.svg License: Public domain Contributors: Own work Original artist: Adrignola• File:Absolute_difference.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/47/Absolute_difference.svg License: CC

BY-SA 3.0 Contributors: Own work Original artist: Dcoetzee• File:AdjunctionSpace-01.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/9e/AdjunctionSpace-01.svg License: Pub-

lic domain Contributors: ? Original artist: ?• File:Ambox_important.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b4/Ambox_important.svg License: Public do-

main Contributors: Own work, based off of Image:Ambox scales.svg Original artist: Dsmurat (talk · contribs)• File:Animated_Sierpinski_carpet.gif Source: https://upload.wikimedia.org/wikipedia/commons/2/28/Animated_Sierpinski_carpet.gif

License: CC BY-SA 3.0 Contributors: Own work Original artist: KarocksOrkav• File:Animated_construction_of_Sierpinski_Triangle.gif Source: https://upload.wikimedia.org/wikipedia/commons/7/74/Animated_

construction_of_Sierpinski_Triangle.gif License: CC BY-SA 3.0 Contributors: Transferred from en.wikipediaOriginal artist: Original uploader was Dino at en.wikipedia

• File:Arrowhead_curve_1_through_6.png Source: https://upload.wikimedia.org/wikipedia/commons/a/ae/Arrowhead_curve_1_through_6.png License: CC BY-SA 3.0 Contributors: Own work, created with Mathematica 6 Original artist: Robert Dickau

• File:Bijection_between_vectors_and_points_on_number_line.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/34/Bijection_between_vectors_and_points_on_number_line.svg License: CC BY 3.0 Contributors: Own work Original artist: Stephan Kulla(User:Stephan Kulla)

• File:Bing’s_Dogbone.tiff Source: https://upload.wikimedia.org/wikipedia/en/c/c4/Bing%27s_Dogbone.tiff License: Fair use Con-tributors: Bulletin of the American Mathematical Society Original artist: R.H. Bing

• File:Cantor_dust.png Source: https://upload.wikimedia.org/wikipedia/commons/6/65/Cantor_dust.png License: Copyrighted free useContributors: http://en.wikipedia.org/wiki/Image:Cantor_dust.png Original artist: Solkoll

• File:Cantor_set_in_seven_iterations.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/56/Cantor_set_in_seven_iterations.svg License: Public domain Contributors: From en.wikipedia.org Image:Cantor_set_in_seven_iterations.svg Original artist: 127 “rect” <ahref='//commons.wikimedia.org/wiki/File:W3C_valid.svg' class='image' title='This image is valid SVG'><img alt='This image is validSVG' src='https://upload.wikimedia.org/wikipedia/commons/thumb/6/66/W3C_valid.svg/32px-W3C_valid.svg.png' width='32' height='16'srcset='https://upload.wikimedia.org/wikipedia/commons/thumb/6/66/W3C_valid.svg/48px-W3C_valid.svg.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/6/66/W3C_valid.svg/64px-W3C_valid.svg.png 2x' data-file-width='200' data-file-height='100' /></a>

• File:Cantors_cube.jpg Source: https://upload.wikimedia.org/wikipedia/commons/7/78/Cantors_cube.jpg License: Public domain Con-tributors:

• en:Image:Cantors cube.jpg Original artist: User:Solkoll• File:Cayley_graph_of_F2.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/d2/Cayley_graph_of_F2.svg License: Pub-

lic domain Contributors: ? Original artist: ?• File:Cmglee_Cambridge_Science_Festival_2015_Menger_sponge.jpg Source: https://upload.wikimedia.org/wikipedia/commons/2/

2b/Cmglee_Cambridge_Science_Festival_2015_Menger_sponge.jpg License: CC BY-SA 3.0 Contributors: Own work Original artist:Cmglee

• File:Comb.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/91/Comb.svg License: CC BY-SA 3.0 Contributors: Ownwork Original artist: Adam majewski

210 CHAPTER 78. TYCHONOFF SPACE

• File:Commons-logo.svg Source: https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg License: ? Contributors: ? Origi-nal artist: ?

• File:Coord_system_CA_0.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/69/Coord_system_CA_0.svgLicense: Pub-lic domain Contributors: Own work Original artist: Jorge Stolfi

• File:CubeOcathedronDualPair.svg Source: https://upload.wikimedia.org/wikipedia/commons/2/2c/CubeOcathedronDualPair.svg Li-cense: CC BY-SA 3.0 Contributors: File:CubeOcathedronDualPair.jpg Original artist: Original rendering in JPG-Format: DanRadin

• File:Diameter_of_a_Set.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/fc/Diameter_of_a_Set.svgLicense: CC0Con-tributors: Own work Original artist: Loren Cobb

• File:Double_comb.svg Source: https://upload.wikimedia.org/wikipedia/commons/8/8d/Double_comb.svgLicense: CCBY-SA3.0Con-tributors: own work with help of experts from Maxima CAS team. Thx Original artist: Adam majewski

• File:DunceHatSpace.png Source: https://upload.wikimedia.org/wikipedia/commons/8/80/DunceHatSpace.png License: Public domainContributors: ? Original artist: ?

• File:Dunce_hat_animated.gif Source: https://upload.wikimedia.org/wikipedia/commons/5/54/Dunce_hat_animated.gif License: CCBY 2.5 Contributors: Own Work, thanks to POVRAY Original artist: Claudio Rocchini

• File:E-to-the-i-pi.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/35/E-to-the-i-pi.svg License: CC BY 2.5 Contribu-tors: ? Original artist: ?

• File:Edit-clear.svg Source: https://upload.wikimedia.org/wikipedia/en/f/f2/Edit-clear.svg License: Public domain Contributors: TheTango! Desktop Project. Original artist:The people from the Tango! project. And according to themeta-data in the file, specifically: “Andreas Nilsson, and Jakub Steiner (althoughminimally).”

• File:Epsilon_Umgebung.svg Source: https://upload.wikimedia.org/wikipedia/commons/c/c1/Epsilon_Umgebung.svgLicense: CC-BY-SA-3.0 Contributors:

• Konvergenz.svg Original artist: Konvergenz.svg: Matthias Vogelgesang (Youthenergy)• File:Euclid_Tetrahedron_4.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/60/Euclid_Tetrahedron_4.svgLicense: CC

BY-SA 3.0 Contributors: Own work Original artist: Aldoaldoz• File:Fractal_fern_explained.png Source: https://upload.wikimedia.org/wikipedia/commons/4/4b/Fractal_fern_explained.pngLicense:

Public domain Contributors: Own work Original artist: António Miguel de Campos• File:Hausdorff_space.svg Source: https://upload.wikimedia.org/wikipedia/commons/2/2a/Hausdorff_space.svg License: Copyrighted

free use Contributors: Based on an image by Toby Bartels. See http://tobybartels.name/copyright/. Original artist: Fibonacci• File:Hawaiian_Earrings.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/e4/Hawaiian_Earrings.svg License: Public

domain Contributors: Transferred from en.wikipedia to Commons. Original artist: Gspr at English Wikipedia• File:Hyperbolic_orthogonal_dodecahedral_honeycomb.png Source: https://upload.wikimedia.org/wikipedia/commons/1/11/Hyperbolic_

orthogonal_dodecahedral_honeycomb.png License: Public domain Contributors: ? Original artist: ?• File:Icosahedron.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b7/Icosahedron.svg License: CC-BY-SA-3.0 Con-

tributors: Vectorisation of Image:Icosahedron.jpg Original artist: User:DTR• File:Illustration_of_supremum.svg Source: https://upload.wikimedia.org/wikipedia/commons/8/82/Illustration_of_supremum.svgLi-

cense: CC BY-SA 4.0 Contributors: Own work Original artist: Stephan Kulla (User:Stephan Kulla)• File:Infinite_broom.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/67/Infinite_broom.svg License: CC BY-SA 3.0

Contributors: File:Infinite broom.png Original artist: Derlay• File:KleinBottle-01.png Source: https://upload.wikimedia.org/wikipedia/commons/4/46/KleinBottle-01.png License: Public domain

Contributors: ? Original artist: ?• File:Kleinsche_Flasche.png Source: https://upload.wikimedia.org/wikipedia/commons/b/b9/Kleinsche_Flasche.png License: CC BY-

SA 3.0 Contributors: Own work Original artist: Rutebir• File:Kuratowski_fan.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/91/Kuratowski_fan.svg License: CCBY-SA 4.0

Contributors: Own work Original artist: Gknor• File:Lebesgue_Icon.svg Source: https://upload.wikimedia.org/wikipedia/commons/c/c9/Lebesgue_Icon.svg License: Public domain

Contributors: w:Image:Lebesgue_Icon.svg Original artist: w:User:James pic• File:Linear_subspaces_with_shading.svg Source: https://upload.wikimedia.org/wikipedia/commons/2/2f/Linear_subspaces_with_shading.

svg License: CC BY-SA 3.0 Contributors: Own work (Original text: Own work, based on en::Image:Linearsubspace.svg (by en:User:Jakob.scholbach).)Original artist: Alksentrs at en.wikipedia

• File:Megamenger_Bath.jpg Source: https://upload.wikimedia.org/wikipedia/commons/1/13/Megamenger_Bath.jpg License: CC BY-SA 4.0 Contributors: Own work Original artist: Photograph by Mike Peel (www.mikepeel.net).

• File:Menger-Schwamm-farbig.png Source: https://upload.wikimedia.org/wikipedia/commons/5/52/Menger-Schwamm-farbig.pngLi-cense: CC BY 3.0 Contributors: Own work Original artist: Niabot

• File:Menger4_Coupe.jpg Source: https://upload.wikimedia.org/wikipedia/commons/3/36/Menger4_Coupe.jpg License: GFDL Con-tributors: Own work Original artist: Theon.

• File:Menger_0.PNG Source: https://upload.wikimedia.org/wikipedia/commons/8/82/Menger_0.PNG License: CC-BY-SA-3.0 Con-tributors: ? Original artist: ?

• File:Menger_1.PNG Source: https://upload.wikimedia.org/wikipedia/commons/f/f6/Menger_1.PNG License: CC-BY-SA-3.0 Contrib-utors: ? Original artist: ?

78.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 211

• File:Menger_2.PNG Source: https://upload.wikimedia.org/wikipedia/commons/9/96/Menger_2.PNG License: CC-BY-SA-3.0 Con-tributors: ? Original artist: ?

• File:Menger_3.PNG Source: https://upload.wikimedia.org/wikipedia/commons/c/cc/Menger_3.PNG License: CC-BY-SA-3.0 Contrib-utors: ? Original artist: ?

• File:Menger_4.PNG Source: https://upload.wikimedia.org/wikipedia/commons/c/c0/Menger_4.PNG License: CC-BY-SA-3.0Contrib-utors: ? Original artist: ?

• File:Menger_5.PNG Source: https://upload.wikimedia.org/wikipedia/commons/6/6a/Menger_5.PNG License: CC-BY-SA-3.0 Con-tributors: ? Original artist: ?

• File:Menger_sponge_(Level_0-3).jpg Source: https://upload.wikimedia.org/wikipedia/commons/d/de/Menger_sponge_%28Level_0-3%29.jpg License: Public domain Contributors: ? Original artist: ?

• File:Mengerova_houba.jpg Source: https://upload.wikimedia.org/wikipedia/commons/a/a6/Mengerova_houba.jpg License: CC BY3.0 Contributors: Own work Original artist: Lukax

• File:Merge-arrows.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/52/Merge-arrows.svgLicense: Public domainCon-tributors: ? Original artist: ?

• File:Multigrade_operator_AND.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/74/Multigrade_operator_AND.svgLicense: Public domain Contributors: Own work Original artist: Watchduck (a.k.a. Tilman Piesk)

• File:Niemytzki_disk.png Source: https://upload.wikimedia.org/wikipedia/en/c/c2/Niemytzki_disk.png License: Cc-by-sa-3.0 Contrib-utors: ? Original artist: ?

• File:Number_line_with_x_smaller_than_y.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/77/Number_line_with_x_smaller_than_y.svg License: CC BY 3.0 Contributors: Own work Original artist: Stephan Kulla (User:Stephan Kulla)

• File:Nuvola_apps_edu_mathematics_blue-p.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg License: GPL Contributors: Derivative work from Image:Nuvola apps edu mathematics.png and Image:Nuvolaapps edu mathematics-p.svg Original artist: David Vignoni (original icon); Flamurai (SVG convertion); bayo (color)

• File:POV-Ray-Dodecahedron.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a4/Dodecahedron.svg License: CC-BY-SA-3.0 Contributors: Vectorisation of Image:Dodecahedron.jpg Original artist: User:DTR

• File:Parabolic_coords.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/90/Parabolic_coords.svg License: GFDL Con-tributors: Own work Original artist: Incnis Mrsi

• File:Pinched_torus.jpg Source: https://upload.wikimedia.org/wikipedia/commons/2/2c/Pinched_torus.jpgLicense: Public domainCon-tributors: Own work Original artist: User:Fly by Night at en.wikipedia

• File:PointedSpace-01.png Source: https://upload.wikimedia.org/wikipedia/commons/2/21/PointedSpace-01.png License: Public do-main Contributors: ? Original artist: ?

• File:Polar_concept_introduction.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/79/Polar_concept_introduction.svgLicense: CC BY-SA 1.0 Contributors: Own work Original artist: Kayau

• File:Quadray.gif Source: https://upload.wikimedia.org/wikipedia/commons/9/99/Quadray.gif License: FAL Contributors: Transferredfrom en.wikipedia to Commons by IngerAlHaosului using CommonsHelper. Original artist: Original author and uploader was Kirbyurnerat en.wikipedia

• File:Question_book-new.svg Source: https://upload.wikimedia.org/wikipedia/en/9/99/Question_book-new.svg License: Cc-by-sa-3.0Contributors:Created from scratch in Adobe Illustrator. Based on Image:Question book.png created by User:Equazcion Original artist:Tkgd2007

• File:Random_Sierpinski_Triangle_animation.gif Source: https://upload.wikimedia.org/wikipedia/commons/a/ad/Random_Sierpinski_Triangle_animation.gif License: CC BY-SA 3.0 Contributors: Own work Original artist: Akarpe

• File:Real_number_line.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/d7/Real_number_line.svg License: Public do-main Contributors: http://www.ams.org/tex/type1-fonts.html Original artist: User:Phrood

• File:Real_projective_line.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/4c/Real_projective_line.svg License: Pub-lic domain Contributors: Own work Original artist: Oleg Alexandrov

• File:Repere_espace.png Source: https://upload.wikimedia.org/wikipedia/commons/b/ba/Repere_espace.png License: CC-BY-SA-3.0Contributors: Transferred from fr.wikipedia to Commons by Korrigan using CommonsHelper. Original artist: The original uploader wasHB at French Wikipedia

• File:Root_system_G2.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/42/Root_system_G2.svg License: CC-BY-SA-3.0 Contributors:

• Root-system-G2.png Original artist: Root-system-G2.png: User:Maksim• File:Sierpinski1.png Source: https://upload.wikimedia.org/wikipedia/commons/3/31/Sierpinski1.png License: Public domain Contrib-

utors: self made using own JAVA animation Original artist: António Miguel de Campos• File:Sierpinski_chaos_animated.gif Source: https://upload.wikimedia.org/wikipedia/commons/f/ff/Sierpinski_chaos_animated.gifLi-

cense: Public domain Contributors: Own work Original artist: Zoharby• File:Sierpinski_pyramid.png Source: https://upload.wikimedia.org/wikipedia/commons/b/b4/Sierpinski_pyramid.png License: CC-

BY-SA-3.0 Contributors: ? Original artist: ?• File:Sierpinski_triangle.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/45/Sierpinski_triangle.svg License: CC BY-

SA 3.0 Contributors: ? Original artist: ?• File:Sierpinski_triangle_evolution.svg Source: https://upload.wikimedia.org/wikipedia/commons/0/05/Sierpinski_triangle_evolution.

svg License: Public domain Contributors: Own work Original artist: Wereon

212 CHAPTER 78. TYCHONOFF SPACE

• File:Sierpinski_triangle_evolution_square.svg Source: https://upload.wikimedia.org/wikipedia/commons/c/c9/Sierpinski_triangle_evolution_square.svg License: Public domain Contributors: ? Original artist: ?

• File:Sierpiński_Pyramid_from_Above.PNG Source: https://upload.wikimedia.org/wikipedia/en/5/5b/Sierpi%C5%84ski_Pyramid_from_Above.PNG License: CC-BY-SA-3.0 Contributors: ? Original artist: ?

• File:Simplicial_complex_example.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/50/Simplicial_complex_example.svg License: Public domain Contributors: Own work. Derived from <a href='//commons.wikimedia.org/wiki/File:Simplicial_complex_example.png' class='image'><img alt='Simplicial complex example.png' src='https://upload.wikimedia.org/wikipedia/commons/thumb/5/55/Simplicial_complex_example.png/25px-Simplicial_complex_example.png' width='25' height='24' srcset='https://upload.wikimedia.org/wikipedia/commons/thumb/5/55/Simplicial_complex_example.png/38px-Simplicial_complex_example.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/5/55/Simplicial_complex_example.png/50px-Simplicial_complex_example.png 2x' data-file-width='532'data-file-height='517' /></a> File:Simplicial complex example.png by Trevorgoodchild(en.wp), released under PD-self. Coloured usingInkscape. Original artist: cflm (talk)

• File:Smith-Volterra-Cantor_set.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/98/Smith-Volterra-Cantor_set.svgLicense: Public domain Contributors: This vector image was created with Inkscape. Original artist: User:Inductivleoad

• File:Smith-Volterra_set.png Source: https://upload.wikimedia.org/wikipedia/commons/c/c1/Smith-Volterra_set.pngLicense: CC-BY-SA-3.0 Contributors: Nomachine-readable source provided. Ownwork assumed (based on copyright claims). Original artist: Nomachine-readable author provided. Henry Bottomley~commonswiki assumed (based on copyright claims).

• File:Sorgenfrey_plane.png Source: https://upload.wikimedia.org/wikipedia/en/b/ba/Sorgenfrey_plane.png License: PD Contributors:? Original artist: ?

• File:Sphere_wireframe.svg Source: https://upload.wikimedia.org/wikipedia/commons/0/08/Sphere_wireframe.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Geek3

• File:Text_document_with_red_question_mark.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a4/Text_document_with_red_question_mark.svg License: Public domain Contributors: Created by bdesham with Inkscape; based upon Text-x-generic.svgfrom the Tango project. Original artist: Benjamin D. Esham (bdesham)

• File:Topological_Rose.png Source: https://upload.wikimedia.org/wikipedia/commons/c/cf/Topological_Rose.png License: Public do-main Contributors: Transferred from en.wikipedia; transferred to Commons by User:Ylebru using CommonsHelper. Original artist:Original uploader was Jim.belk at en.wikipedia

• File:Topological_space_examples.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/ff/Topological_space_examples.svgLicense: Public domain Contributors: Own work Original artist: Dcoetzee

• File:Topologist’s_sine_curve.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/d2/Topologist%27s_sine_curve.svgLi-cense: CC-BY-SA-3.0 Contributors: Own work Original artist: Morn the Gorn

• File:Torus_cycles.png Source: https://upload.wikimedia.org/wikipedia/commons/5/54/Torus_cycles.pngLicense: CC-BY-SA-3.0Con-tributors: ? Original artist: ?

• File:Triangle_angles_sum_to_180_degrees.svg Source: https://upload.wikimedia.org/wikipedia/commons/0/02/Triangle_angles_sum_to_180_degrees.svg License: Public domain Contributors: Own work Original artist: Master Uegly

• File:Up2_2_31_t0_E7.svg Source: https://upload.wikimedia.org/wikipedia/commons/0/0e/Up2_2_31_t0_E7.svg License: Public do-main Contributors: Own work Original artist: User:Tomruen

• File:Wedge_of_Two_Circles.png Source: https://upload.wikimedia.org/wikipedia/commons/6/65/Wedge_of_Two_Circles.pngLicense:Public domain Contributors: Transferred from en.wikipedia to Commons by Ylebru using CommonsHelper. Original artist: Jim.belk atEnglish Wikipedia

• File:Wiki_letter_w.svg Source: https://upload.wikimedia.org/wikipedia/en/6/6c/Wiki_letter_w.svg License: Cc-by-sa-3.0 Contributors:? Original artist: ?

• File:Wiki_letter_w_cropped.svg Source: https://upload.wikimedia.org/wikipedia/commons/1/1c/Wiki_letter_w_cropped.svg License:CC-BY-SA-3.0 Contributors:

• Wiki_letter_w.svg Original artist: Wiki_letter_w.svg: Jarkko Piiroinen

78.5.3 Content license• Creative Commons Attribution-Share Alike 3.0