engelking, sieklucki - topology a geometric approach

Sigma Series in Pure Mathematics Volume 4

Ryszard Engelking Karol Sieklucki

Topology A Geometric Approach

Heldermann Verlag Berlin

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Contents

Foreword ........................................................................ vii

Chapter O: Introduction ........................................................ 1 0.1 Set theory ................................................................ 1 0.2 Algebra ................................................................... 4 0.3 Analysis .................................................................. 6 0.4 Geometry ................................................................. 7

Chapter 1: Metric spaces ...................................................... 13 1.1 Concept of a metric space ................................................ 14 1.2 Operations on metric spaces .............................................. 19 1.3 Maps on metric spaces ................................................... 21 1.4 Metric concepts .......................................................... 32 1.5 Convergence and limits ................................................... 36 1.6 Open and closed sets ..................................................... 41 1.7 Connected spaces ........................................................ 51 1.8 Compact spaces .......................................................... 59 1.9 Complete spaces ......................................................... 65 1.10 Metric and topological concepts in Euclidean spaces ...................... 68 1.S Supplements ............................................................. 76 1.P Problems ................................................................ 82

Chapter 2: Polyhedra .......................................................... 86 2.1 Simplices ................................................................ 87 2.2 Simplicial complexes ..................................................... 91 2.3 Polyhedra ................................................................ 94 2.4 Subdivisions ............................................................. 99 2.5 Simplicial maps ......................................................... 102 2.6 Cell complexes .......................................................... 107 2.S Supplements ............................................................ 112 2.P Problems ............................................................... 117

Chapter 3: Homotopy ........................................................ 121 3.1 Extensions of continuous maps .......................................... 122 3.2 Homotopic maps ........................................................ 131 3.3 Fibrations and coverings ................................................ 140 3.4 The fundamental group ................................................. 150 3.S Supplements ............................................................ 169 3.P Problems ............................................................... 173

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Chapter 4: The topology of Euclidean spaces .............................. 177 4 .1 Maps into spheres ....................................................... 177 4.2 Topological invariance of certain properties of sets ....................... 182 4.3 The theory of position .................................................. 186 4.4 Various examples ....................................................... 198 4.S Supplements ............................................................ 206 4.P Problems ............................................................... 208

Chapter 5: Manifolds .................................................... , .... 211 5.1 The concept of a topological manifold ................................... 211 5.2 Orientability of a manifold .............................................. 219 5.3 Pastings and cuttings ................................................... 222 5.4 Classificaton of 1- and 2-dimensional manifolds .......................... 228 5.S Supplements ............................................................ 240 5.P Problems ............................................................... 243

Chapter 6: Metric spaces II .................................................. 246 6.1 Countable products of metric spaces ..................................... 247 6.2 Spaces of maps ......................................................... 254 6.3 Separable spaces ........................................................ 259 6.4 Complete spaces and completions ....................................... 266 6.5 Continua ............................................................... 276 6.6 Absolute retracts and absolute neighbourhood retracts .................. 287 6. 7 The dimension of separable metric spaces ................................ 295 6.8 Dimension in Euclidean spaces .......................................... 304 6.S Supplements ............................................................ 312 6.P Problems ............................................................... 322

Chapter 'I: Topological spaces ............................................... 331 7 .1 The concept of a topological space ...................................... 332 7 .2 Maps on topological spaces .............................................. 342 7 .3 Separation axioms ...................................................... 348 7.4 Operations on topological spaces ........................................ 356 7.5 Compact spaces and compactifications ................................... 373 7 .6 Metrization of topological spaces. Paracompact spaces ................... 391 7 .S Supplements ............................................................ 397 7 .P Problems ............................................................... 406

Bibliography ................................................................... 418

Subject Index .............. .................................................... 419

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Foreword

This book is an introduction to general and to geometric topology. It was the authors' intention to create a book which is as far as possible not reliant on texts from other branches of mathematics. Consequently the extensive Introduction (treated here as Chapter 0) collects together the basic concepts and facts from set theory, algebra, analysis and geometry which are essential to our own development. Nevertheless we do not recommend that the Introduction should be read in advance of the main text; rather it should be made use of as and when the need arises, by way of references from the main text, or via the index of terms.

Chapter 1, which is devoted to the elementary theory of metric spaces, is also of a distinctive character. It includes much material that is presumably known to the reader from courses in mathematical analysis and geometry. We present this material in an orderly fashion so as to have the required conceptual apparatus at our disposal.

In the following chapters we try to progress gradually from spaces close to intuition and with paradigm properties to spaces which are more and more general and abstract. Chapter 2 is dedicated to polyhedra, which are in a sense the simplest spaces to be studied in topology. It includes an account of the geometric and topological properties of simplices, the theory of simplicial complexes and their subdivisions, the theory of simplicial maps and an equivalent account of the theory of polyhedra based on cell complexes. In Chapter 3 we develop homotopy theory, a body of knowledge which is used in almost all branches of modern topology. In that chapter we also consider some theories for which homotopy is the natural tool; thus we consider the problem of extending continuous maps, fi.bration and covering theory, and the problem of lifting continuous maps; at this juncture we also develop the theory of the fundamental group.

Chapter 4 is devoted to the topology of Euclidean spaces; in a sense this is a continuation of Section 1.10. Amongst other things we prove here the classical theorems on the invariance of separation and the invariance of the interior point, we present an elementary introduction to the theory of position and describe a series of examples of sets and mappings which every mathematician ought to know. In Chapter 5 we are concerned with topological manifolds, that is spaces which are closely linked to the Euclidean ones. Particular attention is paid to the 2-dimensional manifolds, or surfaces, and their classification.

Chapter 6 continues Chapter 1 and is devoted to metric spaces. We expand our stock of operations on metric spaces and undertake a detailed analysis of separable spaces, complete spaces and continua. We also study two classes of spaces with a regular structure: absolute retracts and absolute neighbourhood retracts, and introduce and study the concept of dimension. Chapter 7 contains an elementary treatment of general topology. After introducing basic concepts, we consider operations on topological spaces and thoroughly study the class of compact spaces. We close the chapter by considering paracompact spaces and the metrizability of topological spaces.

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In view of the book's intended scope we have tried to avoid excessive generalisation in the earlier sections of our text. Adoption of such an approach leads unavoidably to some repetition. Certain results enunciated for metric spaces could have been proved outright for arbitrary topological spaces. However, we prefer to prove them first in a particular case and then, when we reach Chapter 7, which is devoted to topological spaces, we just give the appropriate references or even repeat the proof. For instance, we prove Tietze's Theorem twice: first for metric spaces and then separately for topological spaces, since the first of these proofs can be carried out with a more modest conceptual apparatus.

Every section is given a two-part number a.b, where a is the chapter number and bis the number of the section in natural order in that chapter. The last two sections of each chapter are reserved for: supplements (with a label of the form a.S) and problems (carrying a label of the form a.P). The supplements contain historical, terminological and bibliographic comments and information about concepts and results for which room could not be allotted in the main body of text as they fall outside the book's scope but nevertheless deserve mention or more thorough discussion. Some of the problems are difficult and serve to encourage the reader to provide his own proofs. However, the exercises placed at the ends of all the sections are of a different character; these are easy (though not computational) and are there to test command of the material. Figures refer to the text, but never vice versa. The captions under the illustrations are sometimes simplified versions of the theorems being illustrated.

Basic results are stated as theorems, assertions, lemmas, corollaries and examples. Assertions are distinguished from theorems by their self-evidence which permits the omission of a proof. Lemmas have ancillary status only. Examples quite apart from their construction frequently contain a proof that the construction yields the appropriate properties. Each of the units mentioned carries a three-part number a.b.c, where a is the chapter number, b the number of the section in the chapter, whilst c is the position number within the section. Units within a supplement have labels of the form a.S.c and problems are labelled a.P.c. The symbol signifies the end of a section of text headed by a three-part number. We place in square brackets reference numbers to other textbooks or monographs listed in the bibliography.

In closing we wish to express our thanks to all those who helped us write and publish this book. We are particularly indebted to K. Krzyzewski and M. Galecki, who contributed very many apt remarks and corrections. J. Lysko's observations helped us to improve the exercises and problem sections.

Warsaw, July 1992 Ryszard Engelking Karol Sieklucki

1

Chapter 0

Introduction

We assume that the reader is familiar with the basic facts and ideas of set theory, algebra, analysis and geometry. Some of these - especially those required for this book - are recalled here in concise form. The current chapter thus also fixes terminology and notation, and suggests background reading.

0.1. Set theory

Sets will usually be denoted by upper case letters, their elements by lower case letters. If the ~lement a belongs to the set A, we write a E A. The set of elements belonging to X which satisfy the predicate rp will be written {x E X: rp(x)}. If every element of the set A is also an element of the set B, we say that A is a subset of B and we write A C B. If moreover A f. B, then we say that A is a proper subset of B. The empty set 0, i.e. the set with no elements, is a subset of every set. The relation C is called inclusion. We say that the sequence of sets Ai, A2, ... is increasing, if An C An+l for n = 1, 2, ... , and is decreasing if An+l C An for n = 1, 2, ...

A map f of the set X into the set Y is written f: X -+ Y; here X is known as the domain set and Y as the codomain. Maps into the set of real numbers are often referred to simply as functions. The image of a set A C X under the map f, that is, the set of elements y E Y for which there exists a E A such that f(a) = y, is denoted by the symbol f(A). The inverse image or preimage of the set BC Y under the map f, that is, the set of those elements x EX for which f(x) EB, is denoted by the symbol r 1(B).

For any two maps f: X-+ Y and g: Y -+ Z their composition or superposition is the map gf: X-+ Z defined by the formula (gf)(x) = g(f(x)), for x E X. If X CY the map i: X-+ Y defined by the formula i(x) = x for x E Xis called the inclusion map of X into Y. In the special case when X = Y this inclusion map is called the identity map and is denoted by the symbol idx, or, simply, id, when there is no danger of misunderstanding. For every map /:X-+ Y and every set ACX the map (!IA):A-+ Y defined by the formula (f IA)(a) = f(a), for a EA, is called the restriction of the map f to the set A. If ACX and /o:A-+ Y, then every map /:X-+ Y for which /IA= fo is called an extension of the map /o to the set X.

A map which assigns distinct values to distinct arguments is called an injective map. A map f: X -+ Y for which f ( X) = Y is said to be onto Y, or is said to be surjective.

2 Chapter 0: Introduction

An injective map f of a set X onto a set Y is said to be biiective or a one-to-one map. A bijective map /: X --+ Y possesses an inverse map 1-1: Y --+ X determined uniquely by either of the conditions 1-1 f = idx, f 1-1 = idy; this map is also bijective.

A family G of bijective maps of a set X onto itself is said to be a transformation group if G contains the identity map, if, for each map in G its inverse is in G, and if, further, the composition of any pair of maps of G is in G. Bijective maps of the finite set of natural numbers 1, 2, ... , n onto itself are called n-element permutations; evidently, they form a transformation group. Transformation groups are particular instances of groups which we shall consider in Section 0.2.

A class K of maps will be called a trans/ ormation category if for every pair of maps in K of the form/: X--+ Y and g: Y--+ Z, K contains the composition gf: X--+ Zand for each map/: X--+ Y of K, the class K contains both of the identity maps idx, idy. Elements of the class K are called morphisms; the sets X and Y for which there is a morphism f: X --+ Y are called the obiects of the category K. Transformation categories are instances of categories. A knowledge of category theory helps to detect connections between various concepts.

When writing down the elements a of a set A we often assume that they are in a bijective correspondence with the indices t running through a fixed set T. The element corresponding to the index t E T will be denoted at and we shall write A = {at heT. When employing sets with distinguished indices, one should check whether the concepts being studied depend upon the way in which the elements correspond to the indices; but we often omit such a check when it is obvious that there is no dependence of this kind. A set with indices running through the set of natural numbers N will be called a sequence and instead of using the notation { an}nEN we shall usually write {an}n=l,2, ... or {an}~=l or {ai,a2, ... }, or even {an} In the case of finite sets we make use of the following modified notation dictated both by tradition and practicality: the finite set whose elements are a1, a2, ... , an with ordering determined by the natural correspondence with the order of the indices 1, 2, ... , n will be called a finite sequence, or n-tuple, denoted (ai,a2, ... ,an), or (a;);=1,2, ... ,n or (a;)j=1. In particular a two element sequence will be called an ordered pair or just a pair. By contrast we will write { ai, a2, ... , an}, or {a; h=1,2, ... ,n or even {a; }j=l when we have in mind only the set whose elements are ai, a2, ... , an. In particular the set with just one element a will be denoted {a}.

In the case of a family of sets (all of which are subsets of a fixed set X) we apply the notational conventions above as for an ordinary set. Let JI = {AthET where At c X for t E T. The set of all elements a E X, for which there exists an index t E T such that a E At, is called the union of the family JI, denoted UteT At, or more briefly U JI. In particular the formula LJ{A:

0.1. Set theory 3

element a in the product XteT At. We often identify the map a with the set {athET where at = a(t) E At for t E T. For each to E T the map Pt0 : XteT At -+ At0 defined by the formula Pto ( {at}) = at0 is called the projection of the Cartesian product onto its tih-factor.

In the particular case of a sequence of sets {An}, where n = 1, 2, ... , we write LJ~=i An for the union, n~=i An for the intersection, and X::O=i An for the product. Finally, for a finite family of sets Ai, A2, . .. , An we write Ui=i A;, or Ai U A2 U ... U An for the union, nf=i A;, or Ai n A2 n ... n An for the intersection, and x;=i A;, or Ai x A2 x ... x An for the product. The product x;=i A;, when A; = A for; = 1, 2, ... , n is called the nth power of the set A and will be written Xn A.

If f: X-+ Y, then the set {(x, y) E Xx Y : y = f (x)} is called the graph of the map f. For each set A= {atheT we may wish to consider the function o: Ax A-+ {O, 1}, known as the Kronecker function or the Kronecker delta, which we write for practical reasons as Ost or o! instead of o (as, at); the function is defined by the formula:

o - { O, when t =F s, for all s, t E T. st - 1, when t = s,

If ft: At -+ Bt for every t E T, then the map XteT ft: XteT At -+ XteT Bt, defined by the formula

( X ft) ({at})= {ft(at)} tET

for {at} E XteT At, is called the product map of the maps {ftheT In the special case of a sequence of maps f n: An -+ Bn, where n = 1, 2, ... , the product is written X::O=i f n Finally, in the case of a finite sequence of maps /;:A; -+ B; where j = 1, 2, ... , n, the product map is written X;=i /;, or Ii x '2 x ... x fn If/; = f for j = 1, 2, ... , n, the product x;=i !; is called the nth power of the map f and is written xn f.

Sets A and B which have empty intersection are called disjoint. The set of all elements of A which do not belong to B is called the difference of A and B and is written A \ B; if B C A, then the difference A \ B is called the complement of B in A. The well known De Morgan's Laws hold: X \ nteT At = LlteT(X \ At) and x \ UteTAt = nteT(X \At) for any family of sets {AtheT in x.

A relation ~ on a set A is called an ordering if it is reflexive, i.e. a ~ a holds for any a E A, transitive, i.e. the conditions a ~ b and b ~ c imply a ~ c, and weakly antisymmetric, i.e. the conditions a ~ b and b ~ a imply a = b. For example the inclusion relation is an ordering on the family of all subsets of a fixed set X. The set A equipped with an ordering relation ~ is called an ordered set. A subset Ao (not necessarily proper) of an ordered set A is called linearly ordered if for any two elements a, b E Ao we have either a ~ b or b ~ a. An element a E A satisfying the condition x ~ a for every element x of the subset Ao of the ordered set A is called an upper bound of Ao in A. We call an element a E A maximal if the conditions x E A and a ~ x imply a = x. The following holds.

0.1.1. THEOREM (Kuratowski, Zorn). If every linearly ordered subset Ao of a non-empty ordered set A has an upper bound in A, then the set A has a maximal element.


An element a of a linearly ordered subset Ao of an ordered set A is called the smallest (largest) element of Ao, when a S x (x S a) for every x E Ao. If every non-empty subset of a linearly ordered set has a smallest element, then we say that the set A is well-ordered, and the ordering on the set is to be a well-ordering. The following holds.

0.1.2. THEOREM (Zermelo). For every set there exists a relation which well-orders it. A relation R on a set A is called an equivalence, if it is reflexive, symmetric, i.e.

such that the condition aRb implies bRA, and transitive. Every equivalence relation R on a set A determines a partition of the set into pairwise disjoint subsets called the equivalence classes of the relation R. Two elements a, b E A belong to the same equivalence class if and only if aRb. The equivalence class of the relation R which contains the element a E A will usually be denoted [a], and any element of this class will be called a representative of the class.

Two sets A, B are called equinumerous, if there exists a bijective map from the set A onto B. The relation of equinumeracy is an equivalence, and the equivalence class of a set A is called its power or cardinal number and is denoted card A. The power of a finite set is equal to the number of its elements. The power of the set of natural numbers is denoted by the symbol No (aleph-zero), and that of the real numbers by the symbol c (the continuum). We say that the cardinal number m = card A is less than or equal to the cardinal number n = card B and we write m S n, if there exists a bijective map from A onto a subset of B. The following fundamental result holds.

0.1.3. THEOREM (Cantor, Bernstein). If m Sn and n Sm, then m = n. A set whose power is not larger than N0 is called countable. If m = cardA and n = cardB, the cardinal number mn = card(A x B) is

called the product, whilst assuming additionally that A n B = 0, the cardinal number m + n = card( AU B) is called the sum of the cardinal numbers m and n. The power of the family of all subsets of A, when card A = m, is denoted by the symbol 21n. It may be proved that 2No = c.

A more thorough discussion of the concepts of set theory may be found in [11].

0.2. Algebra

By a group we understand an arbitrary set G on which is defined an operation of multiplication (assigning to elements g,g1 E G their product gg1 E G) satisfying the associativity condition (viz. g(g1g11 ) = (gg1)g11 for any g,g',g" E G}, and in which there is a distinguished element 1 E G known as its unity (also denoted by the symbols e or E}, satisfying gl = g for every g E G and having the property that for every g E G there exists an inverse element g-1 with gg-1 = 1. One checks that under these circumstances lg= g and g-1g = 1 for g E G, and that the unity of G and the assignment of inverse elements are defined uniquely. The group consisting of only a unity element is called trivial.

O.!. Algebra 5

If multiplication is commutative (viz. gg1 = g1g for all g, g1 E G) the group is called commutative or abelian. In the case of abelian groups additive notation is often used and then the operation of multiplication is termed addition and is denoted by +; the element g + g1 is then called the sum of g and g', the unity element is called the zero, denoted 0, and the inverse element to g is called its negative and is written -g.

A non-empty subset Go of the group G is called a subgroup of G, if gg1 E Go for every g,g' E G0 and g-1 E G0 for every g E G0 . The subgroup Go is thus itself a group under the operation of multiplication in G.

The direct product G x H of the groups G, H is the group obtained by equip-ping the Cartesian product of G and H with an operation of multiplication defined by (g,h)(g',h') = (gg1,hh'), using (1,1) as the unity element, and letting (g,h)- 1 = (g-l,h-l).

Associativity of group multiplication allows us to consider the product of an arbi-trary finite number of elements (without the need of brackets). We make the convention that the empty set of elements has product 1. A subset T of a group G generates the group (and its elements are called generators of the group) if every element of G is a product of elements of T. and/or their inverses. Any equation which has on its left hand side an irreducible product of generators and/or their inverses and on the right hand side the unity, is called a relation of the group. We sometimes extend this term to cover equations which have products of generators and/or their inverses on both sides. A group can be specified by giving its sets of generators and their relations. A group for which there are no relations (other than the trivial equation 1 = 1) is called free.

If a group G has generators fo1, g2, ... , gk} and relations { Ri, R2, ... , R1} and the group H has generators {h1, h2, ... , hm} and relations {S1, S2, ... , Sn}, then the group generated by {gi,g2, ... ,gk, hi,h2, ... ,hm} with relations {R1,R2, ... ,R1,Si,S2, ... , Sn} is called the free product of the groups G and H.

A map rp of a group G into a group H is called a homomorphism if rp(gg') = rp(g)rp(g') for all g, g' E G. To define a homomorphism of a group G into a group H it suffices to specify its values on the generators of G in such a way that the relations of G are carried to relations of H. An injective homomorphism is called a monomorphism; a homomorphism that is onto is called an epimorphism; a bijective homomorphism is called an isomorphism. For a homomorphism rp: G -+ H to be an isomorphism it is necessary and sufficient that there exists an inverse homomorphism (in fact, an isomorphism) rp-1: H-+ G satisfying rprp-1 = idH and rp- 1rp = idG. If there exists an isomorphism of G onto H, then we say that the groups are isomorphic.

A ring is an abelian group R (presented in additive notation) in which a further commutative and associative operation of multiplication is defined and an element 1 "I- 0 (known as the unity) is distinguished which satisfies rl = r for all r E R. It is, moreover, understood that addition and multiplication are connected by the distributivity condition (r + s)t = rt+ st for all r, s, t E R. A ring F in which every element r E F other than 0 possesses an inverse element r-1 such that rr-1 = 1, is called a field.

A linear space over a field F (whose elements are called scalars) is an abelian group V (presented in additive notation, whose elements are referred to as vectors) equipped


with a further operation, that of multiplication of vectors by scala:t:s (which results in vectors). The distributivity conditions (r + s)o: = ro: +so:, r(o: + fJ) = ro: + r(J for r, s E F, a:, (3 E V are assumed; the unity of the field F satisfies lo: = a: for every a: E V; multiplication in the field F and multiplication of vectors by scalars are connected by the condition r( so:) = (rs )a: for r, s E F, a: E V. A subset A of a linear space V over the field R of reals is convex if for every pair of vectors a:, (3 E A and for every real number 0 $ r $ 1 we have (1- r)o: + r(J EA.

More details on the subject of groups, rings and fields may be found in [12]. A Boolean algebra is an arbitrary set A equipped with two operations: addition,

u, and multiplication, n, and has two distinguished elements: zero, V, and unity, /\, moreover to every element a E A, corresponds a complement -a E A, and the following axioms are satisfied: (1) a U b = b U a, (2) a U (b Uc) = (a U b) Uc, (3) a UV = a, (4) a U (-a) = A, (5) an (b Uc) = (an b) U (an c) for all a, b,c E A and also the dual axioms obtained from (1)-(5) by interchanging the symbols U and n and the symbols V and/\. An ordering relation$ may be introduced into a Boolean algebra by taking a $ b whenever an b = a. Two Boolean algebras are isomorphic, when there exists a bijective map from one to the other which preserves the operations U and n.

0.3. Analysis

We assume known the basic properties of the set R of real numbers in respect of the operations of addition and multiplication and the usual order. In particular we recall that if the set AC R has an upper bound then it has a least upper bound known as the supremum written sup A. If the set A does not have an upper bound, we take sup A = oo. The infimum, inf A, of a set A C R is defined similarly.

For every function /:X-+ R the number (or symbol oo) sup/(X) is called its supremum and is written sup f. We define the infimum of the function f, inf f, anal-ogously. If the supremum (or inti.mum) of the function f lies in /(X) we say that the function f achieves its supremum (or infimum).

Let a and b be real numbers or the symbols -oo or +oo and let a $ b. (We conventionally assume that for all real numbers r we have -oo < r < +oo and that -oo < +oo). We fix the following notation

[a,b]={rER: a$r$b}, for a,bER, [a,b)={rER: a$r

0.4. Geometry 7

If a ER, each of the sets [a, +oo) and (a, +oo) is called a half-line with endpoint a; the former is a closed half-line the latter is open. If b ER then each of the sets (-oo, b], (-oo, b) is called a half-line with endpoint b; the former is a closed half-line the latter is open. Evidently we also have (-oo, +oo) = R.

From among the intervals and half-lines we single out the closed interval [O, 1] which we will call the unit interval, to be denoted by the symbol I and the half-line [O,+oo) to be denoted by the symbol R+

We also assume that the reader is familiar with complex numbers; in particular we shall be using the modulus lei = va2 + b2 and the complex conjugate c =a - bi of the complex number c =a+ bi.

We shall be freely making use of the convergence of sequences and series of numer-ical terms and in the examples and supplements we shall avail ourselves of derivatives and integrals.

All the required information on this subject may be found in any text on mathe-matical analysis, for example see [13].

0.4. Geometry

If mis any positive integer, the set of all finite sequences of real numbers with m terms is called m-dimensional Euclidean space and is denoted Rm. The I-dimensional Euclidean space R 1 is also called the Euclidean line. Since a finite sequence consisting of one number can be identified with that number, we may regard the Euclidean line R 1 as being the set of real numbers R; it is in this sense that we talk of the real line R. We also refer to the 2-dimensional Euclidean space R 2 as the Euclidean plane. For convenience we also consider the 0-dimensional Euclidean space R 0 which we take to be a one-element set consisting of the empty sequence of real numbers.

The elements of the Euclidean space Rm are called the points of the space. If x = ( x1, x2, , xm) E Rm, the numbers x1, x2, , xm are called the coordinates of the point x. Let us observe that the superscripts do not denote exponentiation, but are the coordinate indices. The advantages of such a notation become clear when we have to index a sequence of points; we then use the appropriate subscript to denote a point.

Let x = (x1,x2 , ,xm), y = (y1,y2 , . ,ym) E Rm and let r be a real number. We define the points:

( 1 1 2 2 m m) x+y= x +y ,x +y , ... ,x +y , ( 1 2 m) rx = rx , rx , ... , rx .

The point x + y is called the sum of the points x and y while the point rx is called the scalar multiple of the point x by r. We also put -x = (-l)x and instead of x + (-y) we write x - y. The point -x is called the negative of the point x and x - y is called the difference of x and y. The point (0, 0, ... , 0) E Rm is called the origin of the coordinate system of the Euclidean space Rm and is written 0. Instead of U) x, where r =/= 0, we also write ; and call this the quotient of the point x by the number r.

8 Oh.apter 0: Introduction

The following is a direct consequence of the definitions above.

0.4.1. ASSERTION. For all points x, y, z E Rm and all real numbers r, s E R the following equations hold: (1) (x + y) + z = x + (y + z), (2) x+ 0 = x, (3) x + (-x) = 0, (4) x+y=y+x,

(5) (r + s)x = rx + sx, (6) r(x + y) = rx + ry, (7) (8)

lx = x, r(sx) = (rs)x.

In other words, the Euclidean space Rm is a linear space over the field R. When adding a larger number of points in place of the expression x1 + X2 + ... + Xn

n ~n we shall also use the notation L;=l x;. The sum L....;=l r'x;, where x; E Rm and r; E R for j = 1, 2, ... , n, is called the linear combination of the points xi, x2, ... , Xn with coefficients r1 , r2 , , r".

Let x = (x1, x2 , , xm), y = (y1 ,y2 , ,ym) E Rm. The number xy = L::1 x'y' is called the scalar product of x and y. A consequence of the definition is the following.

0.4.2. ASSERTION. For arbitrary x,y,z E Rm and any number r E R we have the equations: (1) x. y = y. x, (2) (x + y) z = x z + y z,

(3) (rx) y = r(x y), (4) x. x > 0 if x # 0.

The properties listed in Assertions 0.4.1 and 0.4.2 permit the development of a calculus of points and the use of rules which from the formal point of view are identical with the rules of arithmetic.

Instead of x x we shall write x2 The number llxll = ~ is called the norm of the point x E Rm. From this definition we have the following.

0.4.3. ASSERTION. For any point x E Rm and any number r ER we have: (1) llxll = 0 if and only if x = O, (2) llrxll = lrlllxll.

A set of the form {(x1,x2 , ,xm) E Rm: a~ x' ~ b for i = 1,2, ... ,m}, where a, b E R and a ~ b will be called an m-dimensional cube and will be denoted [a, b]m. The cube [O, l]m will be called them-dimensional unit cube, denoted Im.

Let c E Rm and r > 0. The set B(c; r) = {x E Rm : llx - ell < r} is called the m-dimensional open ball centred at c with radius r. Replacing < by ~ we obtain the definition of an m-dimensional closed ball B(c;t) centred at c with radius r. Finally, re-placing the inequality by an equation we obtain the definition of an (m-1)-dimensional sphere S(c; r) centred at c with radius r. The balls B(O; 1), B(O; 1) and the sphere S(O; 1) are known, respectively, as the m-dimensional open unit ball, the m-dimensional closed unit ball and the (m - 1)-dimensional unit sphere and will be denoted by, respectively, Bm, .Bm and 5m-1 2-dimensional balls are also known as discs and the 1-dimensional spheres as circles.

A finite sequence of points ao, a1, ... , an E Rm is said to be affinely dependent if there exists numbers r0 , r1 , , r", not all zero, such that Lf=O r; a; = 0 and Lf=O r; = 0. Otherwise the sequence is affinely independent. Evidently, affine independence does

0.4. Geometry 9

not depend on the order in which the points are listed and the property is inherited by every subsequence of the affinely independent sequence.

The following holds.

0.4.4. THEOREM. In Euclidean space Rm every affinely independent set contains at most m + 1 points; moreover, any such set may be extended to an affinely independent set consisting of precisely m + 1 points.

We say that the set A ~ Rm is in general position, if every set consisting of at most m + 1 points of the set A is affinely independent.

If a, b E Rm and a =/:- b, the set of points of the form x = (1 - r)a + rb, where r E R is called a line through a and b. If the line L passes through the distinct points a and b, while the line L' passes through the distinct points a' and b', then we say that L' is parallel to L if b1 - a' = r(b - a) for some number r. It is readily verified that this definition does not depend on the choice of points a, b of L nor on the choice of a', b' on L' and that the relation of parallelism between lines is an equivalence. The equivalence classes of this relation are called directions.

A set H C Rm with the p~operty that for every pair of distinct points of H the line through the two points lies in H, is called an affine subspace. Thus the empty set, single points, all lines and the whole space Rm are affine subspaces in Rm.

0.4.5. ASSERTION. The intersection of any family of affine subspaces in Rm is an affine subspace in Rm.

. For any set A C Rm the intersection of all the affine subspaces containing A is called the_ affine hull of A and will be denoted H(A). 0.4.6. THEOREM. If A C Rm, then H(A) consists of all points x E Rm of the form x = Lf=O ria;, where a; EA, ri ER /or j = 0, 1, ... ,n and Lf=O ,.,- = 1. 0.4.7. THEOREM. Any affine subspace is the hull of some affinely independent subset of points. Any two such subsets have an equal number of elements.

The number, diminished by 1, of points in an affinely independent set whose hull is the affine subspace H is called the dimension of H, denoted dimH. Thus the dimension of the empty set is -1, that of a single point is 0, and that of a line is 1. Affine subspaces of dimension 2 are called planes. We note the following theorem concerning the dimension of an affine subspace.

0.4.8. THEOREM. If H,H' are affine subspaces and H' CH then dimH' ~ dimH; if moreover H =/:- H', then dim H' < dim H.

0.4.9. THEOREM. The points a0 , ai, ... , an form an affinely dependent set, if and only if, they are contained in an affine subspace of dimension less than n.

Affine subspaces of dimension m - 1 in Rm may be described as the solution sets of single linear equations. These are called hyperplanes of Rm.


The following in fact holds.

0.4.10. THEOREM. A hyperplane of Rm is a set of points (x1, x2 , .. , xm) satisfying an equation of the form

m

ao + La;xi = 0, i=l

where the coefficients a1, a2, ... , llm are not all zero. If a hyperplane Hof Rm is described by the equation a 0+ L::1 a;xi = O, then each

of the two sets defined by the inequalities ao + L::,1 a;xi 2: 0 and ao + L::,1 ~;xi ~ 0 is called a closed half-space of the space Rm determined by H; similarly each of the two sets defined by the inequalities a 0 + L::1 a;xi > 0 and a 0 + L::,1 a;xi < 0 is called an open half-space of Rm determined by H. The two closed half-spaces determined by the one hyperplane are said to be complementary.

We say that the finite set of points ao, a1 , . , an E Rm is orthonormal if (a; - ao) (ak - ao) = h;k for;', k = 1, 2, ... , n. In this definition the point a0 plays a distinguished role. The following theorems hold:

0.4.11. THEOREM. Every orthonormal set is affinely independent.

0.4.12. THEOREM. Every affine subspace contains an orthonormal set whose affine hull is the given affine subspace.

0.4.13. THEOREM. In Euclidean space Rm any orthonormal set may be extended to an orthonormal set consisting of m + 1 points.

A set of points of the form x = (1 - r)a + rb where r E R+ is called a half-line with origin a passing through b. The set of points of the form x = (1 - r)a + rb where r E I is called the line segment with endpoints a and b and will be denoted ab. We say that the set A C Rm is convex, if ab C A for all points a, b E A.

0.4.14. EXAMPLE. The following sets are convex: the whole space Rm, every half-space of R"' (open or closed), every affine subspace in Rm, every m-dimensional cube, every m-dimensional ball (open or closed). The (m - 1)-dimensional sphere is not convex for any m > 0.

0.4.15. ASSERTION. The intersection of any family of convex sets in Rm is convex. For any set A C Rm the intersection of all the convex sets of Rm which contain

A is called the convex hull of A, denoted conv A.

0.4.16. THEOREM. If A C R"', then conv A consists of all points x E Rm of the form x = Lf=O ria;, where a; EA, ri 2: 0 for i = 0, 1, ... , n and LJ=O ri = 1.

If an affinely independent set of points a0 , a 1 , , an has affine hull H and x E H, then it may readily be verified that the numbers r0 , r 1, .. , rn appearing in the equation x = Lf=O ria; and satisfying the relation LJ=O ri = 1 (cf. Theorem 0.4.6) are uniquely determined by the point x; we call them the barycentric coordinates of the point x relative to the sequence ao, ai, ... , an.

0.4. Geometry 11

The following is readily verified.

0.4.17. ASSERTION. If relative to the sequence of points ao, a1 , , an the point p has barycentric coordinates r0 , r 1 , , rn and the point q has barycentric coordinates s0 ,

1 n s , ... ,sn, thenp-q=E;=1(r3 -s3 )(a;-ao).

Let H C Rm be the affine hull of an affinely independent sequence of points ao, ai, ... , an. Every map f: H --+ RP with the property

where Lt=O r; = 1 is called an affine transformation. It may readily be checked that this definition does not depend on the choice of the affinely independent sequence whose affine hull is H.

0.4.18. THEOREM. An affine transformation preserves the affine dependence of a set of points and carries any affine subspace into an affine subspace of no greater dimension.

A bijective, affine transformation of an affine subspace onto an affine subspace is called an affine isomorphism. The inverse map of an affine isomorphism is also an affine isomorphism. Two sets A, A' lying, respectively, in the affine subspaces H, H' are called affinely isomorphic if there exists an affine isomorphism of H onto H' which takes A onto A'.

0.4.19. THEOREM. An affine isomorphism preserves affine dependence and indepen-dence, and also the dimension of affine subspaces.

0.4.20. THEOREM. For any two affinely independent sets in Rm, both consisting of the same number of points, there exists an affine transformation of Rm onto itself which takes one set onto the other. For any two affine subspaces of equal dimension lying in Rm there exists an affine transformation of Rm onto itself which takes one affine subspace onto the other.

The m-dimensional projective space pm is the set of equivalence classes on the set Rm+l \ {O} defined by the equivalence relation: a ,..., b whenever b = ra for some real number r. These equivalence classes are regarded as the points of the space pm. The coordinates of a representative of a point x E pm, which are determined up to a constant of proportionality, are called the homogenous coordinates of the point; the coordinates are m + 1 in number and are indexed in order from 0 to m. To simplify notation we will just write x = [ x0 , x1, , xm].

The point [x0,x1, ,xm] E pm will be called proper or improper depending on whether x0 #- O, or x0 = 0. For proper points we may as well assume that x0 = 1. By treating the remaining coordinates x1 , x2, , xm of the proper point [1, x1 , x2 , . , x"'] E P"' as the coordinates of some point of the space Rm, we can say that the space P"' may be obtained from the space R"' by adding the improper points. On the other hand, each improper point [O, x1, x2 , , xm] E pm may be identified with a direction

12 Ch.apter 0: Introduction

in the space Rm, in fact with the direction of the line passing through the points 0 and (x1,x2, ... ,xm).

We mention that each point of the space pm has precisely two representatives on the sphere sm; they are of the form x and -x. We may therefore regard pm as being obtained from the sphere sm after identification of each point with its negative.

More on the geometry of Euclidean and projective spaces may be found in [1].

Chapter 1

Metric spaces

13

One of the most obvious features of the space we live in is its susceptibility to the measurement of distance. This fact lay at the heart of the development of geometry, which was initially the science concerned with making measurements on the earth's surface and with tracing their interdependences. Also, as the physical sciences, par-ticularly astronomy and mechanics, progressed, it was found useful to study the very notion of space as a conceptual framework encompassing various measurements: dis-tances between material points, changes in these distances (that is to say movements), and also dimensions of rigid bodies (rigid in the sense that the distances between their constituent points stay fixed). An examination of the properties that distance possesses in the setting of Euclidean space leads to the observation that some of them are con-sequences of certain others which are particularly simple to state and are intuitively obvious. Many theorems of elementary geometry may be proved using only these basic properties of distance.

In such circumstances, it is natural to introduce a notion of space more general than Euclidean by taking as primitive the distance between a pair of points, and as axioms some of the obvious properties that distance enjoys in Euclidean space. This idea turns out to be fruitful; it leads to the concept of a metric space, which holds an important place in geometry and in geometric topology, and constitutes a point of departure for the further generalizations of the notion of space in general topology.

In Section 1.1 we give the definition of metric spaces, their simplest properties and various examples which are important in geometry, topology and analysis. A number of other examples of metric spaces may also be found in the exercises for that section and separately in the Problems Section. Section 1.2 describes two basic operations on metric spaces: metric subspace and metric product. The introduction of these oper-ations allows us to give further examples of metric spaces in the section. In order to study categories whose objects are metric spaces, we distinguish in Section 1.3 certain classes of maps between these spaces. These comprise non-expansive maps (which do not expand distances), Lipschitz maps, uniformly continuous maps, and continuous maps. The corresponding classes of isomorphisms are: isometries, similarities, uniform homeo-morphisms, homeomorphisms. On the basis of these maps we explain the classification principles for geometric notions underlying what is known as the Erlangen programme.

In the subsequent sections we proceed to a more detailed study of metric concepts. In Section 1.4 we restrict ourselves to those which are strictly metric. We thus introduce the open and closed balls, the diameter of a space, bounded spaces, and bounded maps. Section 1.5 is devoted to the introduction of limits in metric spaces, their basic proper-ties, the characterization of continuous maps by means of limits, and also of pointwise

14 Chapter 1: Metric spaces

convergence and uniform convergence for sequences of maps. In Section 1.6 we discuss the concept of open set, closed set, dense set, boundary set, we give the basic properties of closed and of open sets and also characterize continuous maps by means of open sets, closed sets and neighbourhoods.

In the next part of the Chapter we distinguish certain classes of metric spaces. Thus in Section 1.7 we are concerned with connected spaces, we give their definition, their simplest properties, some examples and also some sufficiency conditions for con-nectedness. Section 1.8 is devoted to compact spaces. We begin with a proof of the Bolzano-Weierstrass Theorem, which is followed by the definition of compact spaces and their simplest properties. Then we prove a number of classic theorems about com-pact spaces (Lebesgue's Lemma, the Borel-Lebesgue Theorem, the Theorems of Cantor, Heine and Weierstrass). Finally in Section 1.9 we examine the class of complete spaces. The section commences with a study of Cauchy sequences; next, we prove Cauchy's Theorem for numeric sequences and give the definition of completeness. Thereafter we give examples and the simplest properties of this concept.

Section 1.10 applies metric and topological notions to the study of Euclidean spaces and their subsets. We will be concerned with the characterization of compact and of complete subspaces of Euclidean spaces, the interdependence of connectedness and convexity, the characterization of regions by means of broken lines and the topological classification of certain convex sets.

We shall return to the discussion of metric spaces in Chapter 6. Here we limit ourselves to information of a basic character merely to gain the conceptual apparatus needed in the succeeding chapters.

1.1. Concept of a metric space

By a metric space we mean an arbitrary set X together with a function p which associates to every pair z, y of elements of X a real number p(x, y) in such a way that the following axioms are obeyed: (Ml) p(x, y) = 0 if and only if x = y, (M2) p(x,y) = p(y,x) for every x,y EX, (M3) p(x,z) :5 p(x,y) + p(y,z) for every x,y,z EX.

The members of the set X are conventionally called points, the function p is known as the metric and the value p(x, y) of the metric corresponding to the points x, y E Xis said to be the distance between these points (see Supplement 1.S.1). We draw attention to the fact that a metric space is a pair (X,p). The same set X may in general support many functions p: Xx X -+ R satisfying the axioms (Ml)-(M3); each of them is said to metrize the set X. If a metric p on the set X is fixed, or its prescription is beyond doubt, then the metric space ( X, p) will be denoted for simplicity by the single symbol x.

Axiom (M2) is put more briefly by saying that the metric is a symmetric function. Axiom (M3) goes by the name of the triangle inequality in view of the obvious geometric

1.1. Concept of a metric space 15

interpretation in the case when the three points are the vertices of a triangle in the Euclidean plane.

The distance between two points in a Euclidean space is always a non-negative number. However, there is no need to assume this in the form of a separate axiom, because of the following.

1.1.1. THEOREM. If (X, p) is a metric space, then p(x, y) ;?: 0 for any pair of points x,yEX.

PROOF. Using the axioms (M2), (M3) and (Ml), in that order, we infer that 1 1

p(x, y) = 2(p(x, y) + p(y, x)) ;?: 2p(x, x) = 0.

Observe that from axiom (M3) follows the next theorem, which may be called the polygon inequality.

y

Fig.I. The triangle inequality (axiom (M3)) and the polygon inequality (Theorem 1.1.2) for n = 5.

1.1.2. THEOREM. If (X, p) is a metric space and xi, x2, ... , Xn E X, then n-1

p(xi.xn) ~ LP(x;,x;+i) j=l

PROOF. The proof is by induction on the number of points. In the case n = 2 the given inequality is obvious. Suppose, that

k-1 p(x1, xk) ~ L p(x;, x;+I)

j=l

for any set of points xi, x2, ... , xk E X where k ;?: 2. Then, if x1, x2, ... , xk+1 E X, we have

p(xi.xk+I) ~ p(xi.xk) + p(xk,Xk+d k-1 k ~ LP(x;,x;+I) +p(xk,Xk+d = LP(x;,x;+i),

j=l j=l which completes the proof.

16 Chapter 1: Metric spaces

We now give some examples of metric spaces.

1.1.3. EXAMPLE. The discrete metric space. This consists of an arbitrary set X and a metric p defined by the formula:

{ 0 for x = y, p(x,y)= 1 for x=fay.

Axioms (Ml) and (M2) are satisfied for obvious reasons. To check the triangle inequality, suppose that p(x, z) > p(x, y) + p(y, z) for some points x, y, z EX. Then it must be the case that p(x, z) = 1 and p(x, y) = p(y, z) = 0, so that x =fa z and x = y = z, which is a contradiction.

The metric defined above is called the zero-one metric or the discrete metric on the set X.

1.1.4. EXAMPLE. The real line R. This is the space consisting of the set R of real numbers with metric defined by the formula p(x, y) = Ix - YI for x, y E R. The axioms (Ml) and (M2) are obviously satisfied. The triangle inequality follows from the well-known property of the modulus function for real numbers, thus

p(x,z) =Ix- zl = l(x -y) + (y- z)I:::; Ix -yl + IY - zl = p(x,y) + p(y,z) for any three real numbers x,y, z ER.

1.1.5. EXAMPLE. The m-dimensional Euclidean space Rm. This is the space whose points are m-tuples of real numbers, distance between the points x and y being defined by the formula p(x,y) = llx - Yll It follows immediately from Assertion 0.4.3 that the axioms (Ml) and (M2) are obeyed. To verify the triangle inequality we first prove an inequality connecting the scalar product and the norm of points of the space Rm.

Let a, b E Rm with a =fa 0. Then

Checking the trivial case a = 0 separately, we obtain the inequality

known as the Cauchy-Schwartz inequality. It is obviously equivalent to the inequality la bl $ llall llbll from which it follows that a b $ llall llbll Using this inequality we have iia+bll 2 = (a+b)(a+b) = iiall 2 +2(ab)+llbll 2 $ llall 2 +2llallllbll+llbll 2 = (llall+llbll) 2 We thus obtain the inequality

Ila+ bll $ ii all + llbll known as Minkowski's inequality.

Substituting into Minkowski's inequality a = x - y and b = y - z we obtain a+ b = x - z and so llx - zll $ llx - Yll + llY - zll or p(x, z) $ p(x, y) + p(y, z). 1.1.6. EXAMPLE. The sphere sm-l with angular metric. For any pair of points x, y of the (m - 1)-dimensional sphere sm-l we have by the Cauchy-Schwartz inequality (of

1.1. Concept of a metric space 17

Example 1.1.5) that (x y) 2 :5 llxll 2llYll 2 = 1. It follows that there is exactly one number p(x, y) satisfying 0 :5 p(x, y) :5 "Ir and cos p(x, y) = x y. We show that p is a metric on sm-l.

The condition P(x, y) = 0 is obviously equivalent to the equation x y = 1. Since llx - Yll 2 = llxll 2 - 2x Y + llYll 2 = 2 - 2x y, we have x y = 1 if and only if x = y. This proves that axiom (Ml) is satisfied. Axiom (M2) follows from the commutativity of the scalar product.

To prove the triangle inequality consider the points x, y, z E sm-l and let cos a= x y, cosb = y z, cosc = z x. Substituting p = !(-a+ b + c), q = !(a - b + c), r = ! (a + b - c), s = ! (a+ b + c) and applying some well-known trigonometric formulas we have:

4 sin p sin q sin r sins = 1 + 2 cos a cos b cos c - cos2 a - cos2 b - cos2 c, thus 4sinpsinqsinrsins = (1-cos2 c)(l - cos2 b) - (cos a - cosbcosc)2

= (1 - (x z) 2)(1 - (y z) 2) - ((x y) - (x z)(y z)) 2 = llx - (x z)zll 2 llY - (y z)zll 2 - ((x - (x z)z) (y - (y z)z)) 2

Using the Cauchy-Schwartz inequality for the points x - (x z)z, y - (y z)z we deduce that. sinpsinqsinrsins ~ 0. But 0 :5 a,b,c :5 "Ir, so -!"Ir :5 p,q,r :5 "Ir, 0 :5 s :5 i"ll". Now p + q = c, q + r = a, r + p = b and a, b, c ~ 0, so at most one of the numbers p, q, r can be negative.

IT even one of them were negative, then, since the other two in sum do not exceed "Ir and p + q + r = s, we would have 0 :5 s < "Ir, whence sins ~ 0. Thus among the numbers p, q, r, s exactly one would have negative sine, contradicting what we proved about their sines having a non-negative product. Thus p, q, r ~ 0.

In particular from r ~ 0 we obtain c :5 a+b, that is p(x, z) :5 p(x, y)+p(y, z), which completes the proof of the triangle inequality. The obvious geometric interpretation of the formula cos p(x, y) = x y when llxll = llYll = 1 suggests the name of angular metric on the sphere sm-1 .

1.1. 7. EXAMPLE. The m-dimensional proiective space pm. Following the remark of Section 0.4 the projective space pm may be regarded as being the sphere sm in which every pair of points x, -x has been identified. This naturally permits the introduction of a metric on pm by means of the angular metric p described in Example 1.1.6.

It is obvious that for any pair of points x, y E sm there is exactly one number u(x,y) satisfying the conditions 0 :5 u(x,y) :5 !"Ir, cosu(x,y) =Ix YI We then have

( ) _ {p(x,y), if 0 :5 p(x,y) :5 !"Ir, u x,y - "Ir - p(x,y), if !"Ir :5 p(x,y) :5 "Ir.

It follows that u(x, y) = 0 if and only if x = y or x = -y and also that u(x, y) = u(y, x).

Put = {x,

-x,

if u(x,y) = p(x,y), if u(x, y) = "Ir - p(x, y);

{ z, ~= -z,

if u(y, z) = p(y, z), if u(y, z) = "Ir - P(y, z).

18 Oh.apter 1: Metric spaces

Then u(x, y) = p(e, y), u(y, z) = P(y, ~). By the triangle inequality for the metric p we have P(e.~) $ p(e,y) +p(y,~) = u(x,y) + u(y,z). Since of course u(x,z) $ P(e.~) we have u(x, z) $ u(x, y) + u(y, z).

Letting p([x], [y]) = u(x, y) for x, y E sm we obtain a metric p on pm, where [x] denotes the equivalence class of x under the relation identifying x with -x.

1.1.8. EXAMPLE. The Hilbert space Rw. This is the space whose points are the infinite sequences of real numbers x = {x1,x2, ... } for which E:1 (x1) 2 converges and the distance between the points x = { x1, x2, ... } and y = {y1, y2, ... } is defined by the formula

00

p(x, y) = L)x' - yi)2. i=l

Note first that the series appearing under the square root sign is convergent. This follows from the inequalities O $ (x' - y1) 2 = (x')2 - 2x1yi + (y1) 2 $ 2((x1)2 + (y') 2) for i = 1, 2, ... and the fact that the series E:i (x')2 and E:i (y')2 are assumed convergent.

Checking axioms (Ml) and (M2) presents no difficulty. To prove the triangle inequality, suppose that x = {x1, x2, ... }, y = {y1, y2, ... }, z = {z1, z2, ... } are points of the space Rw and form= 1,2, ... take Xm = (x1,x2, . ,xm), Ym = (y1,y2, ... ,ym), Zm = ( z1 , z2, ... , zm). Then, by the triangle inequality in the space Rm obtained in Example 1.1.5 we have Pm(Xm, Zm) $ Pm(Xm, Ym) + Pm(Ym, Zm), where Pm denotes the metric in the space Rm form = 1,2, ... Taking limits over m we obtain p(x,z) $ p(x, y) + p(y, z). 1.1.9. EXAMPLE. The space of maps. Suppose Xis a non-empty set and Y is a metric space with the property that sup{p(y', y") : y', y 11 E Y} < oo. Consider the set P of all maps/: X--+ Y. In P define the distance between two points f and g by the formula

p(f,g) = sup{p(f(x),g(x)): x EX}.

Observe that from the assumption about the space Y it follows that p(f,g) < oo for any two maps f,g E P.

To check that (P,p) is a metric space it is enough to verify the triangle in-equality, since the axioms (Ml) and (M2) are obviously satisfied. Suppose there-fore that f,g,h E P. From the triangle inequality in the space Y we have that p(f(x), h(x)) $ p(f(x), g(x)) + p(g(x), h(x)) for each member x E X. It follows that

p(f(x), h(x)) $ sup{p(/(x), g(x)) + p(g(x), h(x)) : x E X} $ sup{p(/(x), g(x)) : x E X} + sup{p(g(x), h(x)) : x E X} = p(f,g) + p(g,h), for each x EX.

Hence p(f, h) = sup{p(/(x), h(x)) : x EX} $ p(f, g) + p(g, h).

1.e. Operations on metric spaces 19

Exercises

a) For i = 1, 2, 3 give an example of a function P; which associates to each pair from a three-element set X a real number in such a way that axiom (M;) is not satisfied while the other two axioms for a metric are.

b) Show that the axiom system (Ml), (M2), (M3) is equivalent to the axiom system consisting of (Ml) and (M31), where (M31) p(z, .:z:) :'.5 p(x, y) + p(y, z) for every x, y, z E X.

c) Let sw = {.:z: = {.:z:1,.:z:2, ... } E Rw : z:::,1(.:z:1) 2 = l}. Examine whether the function p defined by the conditions: 0 :'.5 p(x,y) :'.5 11", cosP(.:z:,y) = z:::,1 z1y1 for

{ 12} {12} sw t" x = .:z; , .:z; , . . . , y = y , y , . . . E is a me nc.

1.2. Operations on metric spaces

We now pass to a discussion of certain operations which will allow us to expand on the number of examples of metric spaces.

If (X,p) is a metric space and ACX, then taking PA(x,y) = p(x,y) for x,y EA we obtain a function which is obviously a metric on A. The pair (A, PA) is then called a metric subspace of the space (X,p). If when referring to the space (X, p) we omit the symbol p then we say that A is a metric subspace, or more briefly a subspace, of the space X, meaning to say that the metric PA defined above is used to metrize A. We may sometimes come across situations, where for practical or traditional reasons one speaks of subsets rather than metric subspaces of X. To avoid possible confusion and also unnecessary formality we agree the convention that whenever concepts which refer to the set A c X are metric in character, we tacitly treat A as a metric subspace of the space X.

1.2.1. EXAMPLE. Metric subspaces of the space Rm. The real intervals and half-lines introduced in Section 0.3 may be regarded as metric subspaces of the real line R. Similarly the m-dimensional cubes and balls, the (m - 1)-dimensional spheres, affine subspaces, half-spaces, half-lines and line segments introduced in Section 0.4 may be regarded as metric subspaces of the Euclidean space Rm.

1.2.2. EXAMPLE. The Hilbert cube 1w. This is the metric subspace of the Hilbert space R w defined by the formula

[w = {{.:z:l,.:z;2, ... } E Rw: O ::=; .:z:1 :'.51/i for i = 1,2, ... }. Since 0 ::=; (.:z:1) 2 ::=; (l/i)2 for i = 1, 2, ... and the series z:::,1 (1/i)2 is convergent, we do indeed have JW C R w.

Suppose given a finite sequence of metric spaces (X1, Pi) for i = 1, 2, ... , m. We may define on the set x = x:1 x, a metric p by means of the formula

m

p(x,y) = L:>Hz1,Y1), i=l

where .:z; = (.:z:i,z2, ... ,zm), y = (y1,y2, ... 1 Ym) EX.


Certainly the axioms (Ml) and (M2) follow directly from the respective axioms applied to the metric Pi for i = 1, 2, ... , m. To prove the triangle inequality suppose that the points x = (xi,x2, ... ,xm), y = (y1,y2, ... ,ym), z = (z1,z2, ... ,zm) lie in X; put ai = Pi(xi,Yi), bi= Pi(Yi,zi), ci = Pi(xi,zi) for i = 1,2, ... ,m and then consider in the Euclidean space Rm the points a = (a1 , a2, ... , am), b = (b1, b2, ... , bm), c = ( c1' c2' ... ' cm). From the triangle inequality in the space (xi' Pi) we obtain Ci ~ ai + bi for i = 1, 2, ... , m, hence JlcJI ~ Jla + bll Using the Minkowski inequality proved in Example 1.1.5 we thus have p(x, z) = llcll ~ Ila+ bll ~ llall + JlbJI = p(x, y) + p(y, z).

The set X together with the metric p defined above is called the metric product of the spaces (Xi,Pi) for i = 1,2, ... ,m and we write (X,p) = (X1,P1) x (X2,p2) x ... x (Xm,Pm), or just X = X1 X X2 X X Xm. We shall also use the brief notation (X,p) = X:)Xi,Pi) for the metric product, or simply X = x:1 Xi.

Note now that the m-dimensional Euclidean space Rm may be regarded as the metric product of m copies of the real line. Similarly, the m-dimensional cube Im may be treated as a metric product of m copies of the unit interval. We also see that the following is obvious.

1.2.3. ASSERTION. If Ai is a metric subspace of metric space Xi for i = 1, 2, ... , m, then x:1 Ai is a metric subspace of the space x:1 Xi.

In future we shall wish to make use of the following estimate for the distance in the metric product.

1.2.4. LEMMA. If (X,p) = x:1(Xi,Pi), x = (x1,x2, ... ,xm), and y = (yi,y2, ... ,ym), then

max{pi(Xi,Yi) : i = 1, 2, ... ,m} ~ p(x,y) ~ vmmax{pi(Xi,Yi) : i = 1, 2, ... , m}. PROOF. It is enough to prove that for i = 1, 2, ... , m the inequalities

m

pHxi,Yi) ~ L p~(xi, Yi) ~ mmax{p;(xi,Yi) : i = 1, 2, ... , m}, i=l

hold. The one on the left follows from the non-negativity of the summands, that on the right from the definition of the max function.

Exercises

a) Suppose that (Xi, Pi) is a metric space for i = 1, 2, ... , m and let X = x:1 Xi Show that the function p which associates with every pair of points x = (xi, x2, ... , xm) and Y = (Y1, Y2, ... , Ym) of the set X the number p(x, y) = L:~1 Pi(xi, Yi) is a metric on x.

b) Suppose that (Xi, pi) is a metric space for i = 1, 2, ... , m and let X = x:1 Xi. Show that the function p which associates with every pair of points x = (xi, x2, ... , xm) and y = (Y1,Y2, ... ,ym) of the set X the number p(x,y) = max{pi(xi,Yi) : i = 1, 2, ... ,m} is a metric on X.

1.9. Maps on metric spaces 21

c) Suppose X = Uj=1 X; with X; n X1r. = {xo} for j -:/= k and suppose P; is a metric on X; for j = 1, 2, ... , n. Show that the function p defined by the formula:

p(x, y) = { P;(x, y), P;(x, xo) + P1r.(xo, y),

if x,y EX;, if x E X1., y E X1r., j -:/= k,

is a metric on X.

1.3. Maps on metric spaces

In this section we distinguish certain classes of maps on metric spaces which will enable us to treat metric spaces as the objects of certain categories. Suppose that X and Y are metric spaces and f: X --+ Y. We shall denote the metrics on the spaces X and Y by the same symbol p on the understanding however that the symbol has a different meaning when applied to the points of the space X than when it is applied to the points of the space Y.

We say that the map f is non-expansive when p(f(x), f(x')) :::; p(x, x') for every pair of points x, x1 EX.

.

1.3.1. EXAMPLE. Inclusion of a metric subspace. Let A be a metric subspace of a metric space X. The map iA: A--+ X defined by the formula iA(a) =a for a EA is called the inclusion map of the subspace A into the space X. From the definition of the metric on a subspace it follows immediately that inclusion is non-expansive.

1.3.2. EXAMPLE. Projection of the metric product onto a factor. Let the space X be the metric product of the metric spaces X1, X2, ... , Xm. For i = 1, 2, ... , m define the map Pi: x --+ xi by the formula Pi(xi, x2, ... , Xm) = Xi for (xi, X2, ... , Xm) E x:1 xi It follows from Lemma 1.2.4 that for i = 1, 2,. . ., m we have P(Pi(x), Pi(x')) :::; p(x, x'); the map Pi for i = 1, 2, ... , m, which we call the projection of the product x:1 Xi onto its ith._factor, is thus a non-expansive map.

--------------------1 y XxY

(x',y') I I I I I I I I I I I

x' e(x',x")

(x",y") I I I I I I

x" x

Fig.2. Projection of the metric product Xx Y onto the factor X is non-expansive (Example 1.3.2).


1.3.3. EXAMPLE. Orthogonal projection of the space Rm onto an affine subspace. Let H be the affine "hull of an affinely independent set of points ao, ai, ... , an E Rm and let x E Rm. Any point y E H such that (x - y) (p - q) = 0 for any p,q E H is called an orthogonal projection of the point x onto H. We prove below the existence and uniqueness of the orthogonal projection. By Assertion 0.4.17 it follows that a point y EH is an orthogonal projection of the point x onto H ifand only if(x-y) (a;-ao) = 0 for j = 1,2, ... ,n.

To prove the existence of the orthogonal projection y of the point x into H re-place (using Theorem 0.4.12) the set of points ao, ai, ... , an by an orthonormal set b0 , bi, ... , bn whose hull is also H. Let ri = (x - bo) (b; - bo) for j = 1, 2, ... , n, r0 = 1- Lf=l ri. Putting y = Lt=O rib; E H we have x - y = (x - bo) + (bo - Lf=O rib;) = (x - bo) - Lf=l ri(b; - bo). Hence (x - y) (bk - bo) = rk - Ej=l ric;k = rk - rk = 0 for k = 1, 2, ... , n. Hence y is indeed an orthogonal projection of x onto H.

To prove uniqueness of this projection observe that if y, y E H and ( x - y) ( ao -a;) = 0 and (x-y) (ao-a;) = 0 for j = 1, 2, ... , n, then by subtracting the equations we have (y-y) (a0 -a;) = 0 for j = 1,2, ... ,n. If y = Lt=O ria; and y = E.i=o ;:ia; where E.i=o ri = E.i=o ;:i = 1 then by Assertion 0.4.17 we have y-y = Ej=l (ri -ri)(a;-a0 ). Hence Jly - yJ1 2 = Ej=1(ri - ;:i)(y - y) (a; - a0 ) = 0, so that y - y = 0 or y = y.

We now prove that if y and y denote respectively the orthogonal projections of the points x and x onto H, then p(y,y) ~ p(x,x). For, ify = Ej=0 ria;, y = Ej=0 ria; where E.i=o ri = E.i=o ;:i = 1, then by Assertion 0.4.17 we have y - y = Ej=1 (ri -;:i)(a; - ao), hence (x - y) (y - y) = 0 and (x - y) (y - y) = 0 and so ((x - x) - (y -y)) . (y - y) = 0. From this it follows that llx - xll 2 = II (x - x) - (y - y) 11 2 + llY - Yll 2 and so llY - yJl 2 ~ IJx - xll 2, or JIY - yJI ~ Jlx - xii 1.3.4. EXAMPLE. The norm. The map 11: Rm -+ R defined by 11(x) = llxll is non-expansive. For, if x,x' E Rm, then Jlxll = Jlx' + (x - x')ll ~ llx'll + llx - x'll and Jlx'JI = llx + (x' - x)JI ~ llxll + llx' - xii, hence lllxll - Jlx'lll ~ llx- x'll 1.3.5. EXAMPLE. The map p: sm -+ pm. Associate with each point x of the sphere sm its equivalence class p(x) = [x] in the projective space pm, This is a non-expansive map in the sense of the metrics p and p defined in Examples 1.1.6 and 1.1.7, since p([x], [y]) = p(x, y) when 0 ~ p(x, y) ~ 111" and p([x], [y]) = 11" - p(x, y) when 111" ~ p(x, y) ~ 11". Thus in both cases p([x],[y]) ~ p(x,y).

It is obvious that the identity map is non-expansive and that the composition of two non-expansive maps is itself non-expansive.

If there exists a constant c ~ 0 with the property that p(f(x), f(x')) ~ cp(x, x1) for any pair of points x, x' E X, then f is said to be a Lipschitz map with constant c. Obviously if a map is non-expansive then it is a Lipschitz map with constant 1. The Lipschitz maps thus form a wider class than the non-expansive maps. The composition of two Lipschitz maps with constants c and c' is a Lipschitz map with constant cc'. (See also the Supplements 1.S.7 and 1.S.8).

1.3.6. EXAMPLE. Every real differentiable function f: R -+ R which has derivative bounded by c is a Lipschitz map with constant c. For, by the Mean Value Theorem we


deduce that for any two points x, x' ER there is a point such that

lf(x) - f(x')I = l!'Wllx - x'I $ clx - x'I

1.3.7. EXAMPLE. The metric. The metric p of a metric space (X,p) may be treated as a map p: X x X -+ R. It is then a Lipschitz map with constant v'2. Indeed, if x, x1, y, y1 E X then using Theorem 1.1.2 we have p(x, x') $ p(x, y) + p(y, y1) + p(y', x') and p(y, y') $ p(y, x) + p(x, x') + p(x', y'). Thus

lp(x,x') - p(y,y')I $ p(x,y) + p(x1,y1) $ v'2VP2(x,y) + p2(x',y').

1.3.8. EXAMPLE. Addition of points. The operation of addition for points in the Eu-clidean space Rm may be viewed as a map Rm x Rm -+ Rm. It is then a Lipschitz map with constant y'2. For, if x,x1,y,y1 E Rm, then

p(x + y,x' + y1) = ll(x + y) - (x' + y')ll = ll(x - x') + (y - y')ll . S llx - x'll + llY - Y1ll $ v'2Jllx - x'll 2 + llY - Y'll 2

1.3.9. EXAMPLE. Every affine map is a Lipschitz map. From Theorem 0.4.4 it follows that every affine map/: H-+ RP where His an affine subspace of Rm may be extended to an affine map of the space Rm into RP. It suffices to examine maps of this type.

If a map /:Rm -+RP is affine and we take eo = 0, ei = (of, ol, ... 'ot) E Rm for i = 1, 2, ... , m, then for each x = (x1, x2, ... , xm) E Rm we have

m m

x = (1 - L xi)eo + L xiei, i=l i=l

hence m m m

f(x) = (1- L:xi)f(eo) + L:xif(ei) = f(eo) + L:xi(f(ei) - f(eo)). i=l i=l i=l

If - (-1 -2 -m) th moreover x = x , x , ... , x , en m

11/(x) - f(x)ll =II L(xi - xi)(f(ei) - f(eo))ll i=l

m m

$ L lxi - xilllf(ei) - f(eo)ll $ c L lxi - xii i=l i=l

where c = max{llf(ei) - f(eo)ll : i = 1, 2, ... , m}. Applying Lemma 1.2.4 we obtain

11/(x) - f(x)ll $ cmmax{lxi - xii: i = 1, 2, ... , m} $ cmllx - xii

We now present a class of metric maps more general than the Lipschitz maps. We shall say that a map f is uniformly continuous if for every positive real number E there is a positive real number 6 such that if x, x' E X and p(x, x') < 6 then p(f(x), f(x')) < E.


Every Lipschitz map is uniformly continuous, for if c =/= 0 is its constant then for given E > 0 it is enough to take 6 = E/c.

1.3.10. EXAMPLE. Any map f defined on a discrete metric space X is uniformly con-tinuous. Evidently, for any E > 0 it is enough to take 6 = 1. For if p(x, x') < 6 with x, x1 EX then x = x1 and so f(x) = f(x') and so p(f(x), f(x')) = 0 < E. It easily follows that there are uniformly continuous maps which are not Lipschitz for any constant c.

1.3.11. THEOREM. The composition of two uniformly continuous maps is uniformly continuous.

PROOF. If the maps f: X --+ Y and g: Y --+ Z are uniformly continuous then for every positive real number E there is a positive real number 6 such that if y, y1 E Y and p(y,y') < 6 then p(g(y),g(y')) < E. Corresponding to 6 there is a real number 1J > 0 such that if x, x' E X and p(x, x') < 1J then p(f(x), f(x')) < 6. Hence if x, x' E X and p(x,x') < 11, we have p(gf(x),gf(x')) < E.

We now expand the class of uniformly continuous maps as follows. We say that a map f is continuous at the point x E X if for every positive real number E there is a positive real number 6 such that if x' E X and p(x, x') < 6 then p(f(x), f(x')) < E. A map /: X--+ Y that is continuous at every point of the space X is called continuous (see Supplement 1.S.9}. It follows immediately from this definition that every uniformly continuous map is continuous.

1.3.12. EXAMPLE. Scaling points by reals. The operation of scaling a point of the Euclidean space Rm by a real number may be viewed as a map R x Rm --+ Rm. This map is continuous though not uniformly continuous. Observe that if r, r1 E R and x,x' E Rm, then

p(rx, r1 x') = llrx - r' x'll = llr(x - x') + (r - r')x - (r - r') (x - x') II ~ lrlllx - x'll +Ir - r'lllxll +Ir - r'lllx - x'll

Thus if E > 0, then choosing 6 so that 6(lrl + llxll) < lE and 62 < lE we infer that if y'(r - r'} 2 + llx - x'll 2 < 6, then Ir - r'I < 6 and llx - x'll < 6, and so p(rx,r'x') ~ 6(lrl + llxll) + 62 < E. The scaling operation is thus continuous.

On the other hand for any positive number 6 and fixed point p E Rm with p =/= 0 we may taker= 1/6, r1 = 1/6 + 6/2, x = rp, x' = r'p, then p(rx,r1x1) = llrx - r'x'll = (r'2 - r 2}llPll = (1 + i62}11pll > llPll Thus the scaling operation is not uniformly continuous.

1.3.13. EXAMPLE. The scalar product. The scalar product of points in the Euclidean space Rm may be viewed as a map Rm x Rm--+ R. It is continuous but not uniformly. This may be checked by an argument analogous to that of the last Example.

By an argument similar to the proof of Theorem 1.3.11 we obtain

1.3.14. THEOREM. If a map f: X --+ Y is continuous at the point xo E X and the map g: Y --+ Z is continuous at the point Yo = f(xo), then the composition gf: X --+ Z is continuous at the point xo.


In the discussion above we successively picked out more and more general classes of maps: non-expansive, Lipschitz, uniformly continuous and continuous. Metric spaces taken as objects form a category with each of these classes of maps as morphisms. We now study the isomorphisms in these categories.

A map/: X--+ Y which is non-expansive, bijective and has an inverse 1-1 : Y--+ X which is non-expansive is called an isometric map or simply an isometry. An isometry f: X--+ Y is thus a map of a space X onto a space Y characterized by the condition

p(f(x), f(x')) = p(x,x') for any two points x,x' E X. The isometries of a fixed metric space onto itself form a transformation group.

1.3.15. EXAMPLE. Translations. Fix a point a E Rm and define a map ta: Rm--+ Rm by the formula ta(x) = x +a for x E Rm. Maps of this form are called translations. These are of course isometric, since p(ta(x), ta(x')) = Jlx +a - x' - aJI = llx - x'JI = p(x, x') for any points x,x' E Rm.

Note moreover that to= i~, tbta = ta+b t;;1 =La for every a,b E Rm. Trans-lations thus form an abelian subgroup of the group of isometries of the space Rm. The subgroup of translations has also the additional property that for any two points x, y E Rm there is exactly one translation which takes the point x onto y; this is the translation ty-z

x2

0 x'

Fig.3. The translation ta defined by the formula ta(x) = x +a.

1.3.16. EXAMPLE. Rotations. Fix a real number cp and define a map r\O:R2 --+ R 2 by the formula r\O(x1, x2) = (x1 cos cp - x2 sin cp, x1 sin cp + x2 cos cp) for (x1, x2) E R 2. The map is called an elementary rotation through an angle of cp. We note that we do not define the angle cp as such but only a rotation through an angle cp; however despite this formality the. sense of the definition agrees with the intuitive notion of rotation. Obviously r\O = r1+2irk for k = 0, 1, ...

Every elementary rotation is an isometry. We have

p2 (r \0 ( x), r\O (x')) = ( (x1 - x11 ) cos cp - (x2 - x'2) sin cp )2 + ((x1 - x'1) sin cp + (x2 - x'2) cos cp) 2

=(xi - x'1)2 + (x2 - x'2)2 = p2(x,x'),

26 Okapter 1: Metric spaces

where x = (x1 , x2), x 1 = (x'1 , x'2). Note also that ro = id, r.pr,,, = r.p+'P r;; 1 = r _,,, for each ip, 1/J E R. The elementary rotations thus form an abelian subgroup of the group of isometries of the plane R 2

Let 1 $ i < i $ m. Consider a map/: Rm-+ Rm defined so that if y = f(x) with _ ( 1 2 m) _ ( 1 2 m) th k - k " k ...J. d th d d x - x , x , ... , x , y - y , y , ... , y , en y - x ior r i, J, an e m uce

map taking (xi, xi) E R2 to the point (yi,yi) E R 2 is an elementary rotation. The composition of a finite number of maps of this type (allowing all possible choices of i,i) is called a rotation of the space Rm. For m = 1 the rotations are taken to be just the two maps: identity and the map /(x) = -x for x E R. It is easily checked that every rotation of the space Rm is an isometry fixing the point 0 and that the rotations form an abelian subgroup of the group of isometries of the space Rm.

x'2

0 xi x1 = r cos (a+ g;) =rcosa cosg;-r sin a sinip=x1 cosip-x2sing;, x'2 =r sin (a+g;) = r cos a sing;+ r sin a cosip =x1 sin

1.9. Mapa on metric apacea 27

an isometry since p(a(x), a(x')) = 11- x + x'll = llx - x'll = p(x, x') for x,x' E Rm. Two metric spaces X and Y for which there exists an isometry map of X onto Y

are said to be isometric or congruent. The class of all metric spaces which are isometric to a space Xis called the metric type of the space X. The theory of isometry invariants, that is the theory concerned with those properties of metric spaces which if enjoyed by one space are enjoyed by all spaces isometric to it, is called metric geometry or simply geometry.

1.3.18. EXAMPLE. Any n-dimensional affine subspace of Rm is isometric with the space Rn. Let H C Rm be an n-dimensional affine subspace. By Theorem 0.4.12 we may assume that H is the hull of an orthonormal set of points ao, a1, ... , an. For every point x = Lf=Or;a; EH with Lf=Or; = 1 associate the point f(x) = (r1,r2, ... ,rn) E Rn. To show that f is an isometry consider also the point x = Lf=O r; a; E H with Lf=O r; = 1. By Assertion 0.4.17 we have x - x = E.i=l (rj - r;)(a; - ao), whence

n n

llx - xll 2 = L L(r; - r;)(rk - rk)(a; - ao) (ak - ao) j=l k=ol

n n

= L L(r; - r;)(rk - rk)o;k j=lk=l

n

= L(r; - r;)2 = llf(x) - /(x)ll2. j=l

1.~.19. EXAMPLE. Commutativity of the metric product. Consider the metric spaces (Xi.pi), (X2,p2) and their metric products (X1,pi) x (X2,P2) and (X2,P2) x (Xi.pi). We may map the former product onto the latter by sending the point (xi. x2) to the point (x2, xi). This map is easily seen to be an isometry. From the point of view of metric geometry we may thus identify the two products and in this sense claim that the metric product is commutative.

1.3.20. EXAMPLE. Associativity of the metric product. Consider a sequence of metric spaces (Xi.pi), (X2,p2), ... , (Xm,Pm), a number k such that 1 :5 k :5 m, and the two metric products (X:=i (Xi, Pi)) x (X:k+l (Xi, Pi)) and x:1 (Xi, Pi) We may map the for-mer product onto the latter by sending the point ((xi. x2, ... , xk), (xk+l Xk+l ... , Xm)) to the point (xi. x2, ... , xm) This map is easily seen to be an isometry. From the point of view of metrie geometry we may again identify the two products and in this sense claim that the metric product is associative.

We have the following conclusion from an inductive argument which appeals to Example 1.3.20.

1.3.21. COROLLARY. An arbitrary family of metric spaces which for any pair of members contains their metric product, also contains the metric product of any finite number of its member.

A Lipschitz map /: X -+ Y with constant c which is bijective and has as its inverse 1-1: Y -+ X a Lipschitz map with constant 1/c is called a similarity map with


coefficient c. A similarity/: X - Y with coefficient c is thus a map of the space X onto Y characterized by condition

p(f(x), f(x')) = cp(x,x')

for any pair of points x, x' EX. The similarities of a fixed metric space onto itself form a transformation group.

1.3.22. EXAMPLE. Homotheticity. The map he: Rm - Rm defined by the formula hc(x) = ex for x E Rm is known as a homotheticity with coefficient c, where c is any positive real number. Note that every homotheticity with coefficient c is a similarity with coefficient c. Clearly, p(hc(x),hc(x')) = llcx - cx'll = cllx - x'll = cp(x,x') for all x,x1 E Rm.

Observe moreover that h1 = id, hc2 hc, = hc,c., h-;1 = hi/c for all positive ci, c2, c. It follows that the homotheticities constitute an abelian subgroup of the group of all similarities of the space Rm.

We also see that every similarity of the space Rm may be expressed as the compo-sition of an isometry with a homotheticity. Indeed, if f: Rm - Rm is a similarity with coefficient c then the composition g =hi/cf is a similarity with coefficient (1/c)c = 1, so is an isometry; thus f = hcg where g is an isometry.

0

Fig.5. A homotheticity with coefficient c > 1.

1.3.23. EXAMPLE. The diagonal map. Let X be a metric space. Consider the power Xm X =Xx Xx ... x X where mis any natural number. The map d:X - Xm X defined by the formula d(x) = (x, x, ... , x) E Xm X for x EX is called the diagonal map. It is a similarity with coefficient rm of the set x onto the set t::.. = {(xi, x2, ... 'Xm) E xm x : X1 = X2 = ... = Xm} known as the diagonal of the power xm x. We have

p((x,x, ... ,x), (x',x', ... ,x')) = Jmp2(x,x') = y'mp(x,x1)

for every x, x' E X. (See also Supplement 1.S.12).


Two metric spaces X and Y for which there exists a similarity map from X onto Y are called similar.

1.3.24. EXAMPLE. All non-degenerate closed intervals on the real line are similar. If a< b, the map f: R--+ R defined by f (x) = (b - a)x +a is a similarity with coefficient (b - a) mapping the unit interval I onto the closed interval [a, b]. We may similarly prove that all non-empty open intervals on the real line are similar and also that all non-empty intervals which are left-closed or right-closed are similar.

x

Fig.6. The diagonal map d: X --+ b.. C X x X is a similarity (Example 1.3.23).

The theory of similarity invariants, that is of those properties of metric spaces which if enjoyed by one space are enjoyed by all spaces similar to it is known as similarity geometry.

A uniformly continuous map (respectively, a continuous map) f: X --+ Y which is bijective and has as its inverse f- 1: Y --+ X a map that is likewise uniformly con-tinuous (respectively, continuous) is called a uniform homeomorphism (respectively, a homeomorphism). The uniform homeomorphisms (respectively, the homeomorphisms) of a fixed space onto itself form a transformation group. Two metric spaces X and Y for which there is a uniform homeomorphism (respectively, a homeomorphism) mapping X onto Y are said to be uniformly homeomorphic (respectively, homeomorphic).

1.3.25. EXAMPLE. Let X denote the real line with discrete metric and Y the same set with the usual real line metric. The map id: X --+ Y is continuous (even, uniformly continuous, cf. Example 1.3.10) and bijective but is not a homeomorphism since the inverse map is not continuous. More may be proved, namely that the spaces X and Y are not homeomorphic.

1.3.26. EXAMPLE. Any non-empty open interval and the real line are homeomorphic. By the property given in Example 1.3.24 we may consider the interval (-7r ,'Ir). Then the map f: (-7r,7r) --+ R defined by the formula f(x) = tanx for x E (-7r,7r) is con-tinuous and bijective and the inverse function f- 1 = arc tan is also continuous. The homeomorphism f is however not uniform since the map f is not uniformly continuous. More may be proved, namely that the interval (-7r,7r) (or any other open interval) and the real line R are not uniformly homeomorphic (cf. Corollary 1.9.10).


From Example 1.3.9 we obtain the following.

1.3.27. COROLLARY. Every affine isomorphism is a uniform homeomorphism.

1.3.28. EXAMPLE. Inversion. Observe that if r is any fixed positive real number, then to every point x E Rm\{O} there corresponds just one point i(x) lying on the half-line which has its endpoint at the origin and passes through x, such that lli(x)ll llxll = r2 In fact, if i(x) = (1 - t)O + tx = tx, where t ER+, then since r 2 = lli(x)llllxll = lltxllllxll = t11xll 2, we have t = r 2 /llxll 2 , whence i(x) = (r2 /llxll 2)x for x E Rm\ {O}.

The map i:Rm\{O}-+ Rm\{O} is known as the inversion in the sphere centred at O of radius r. It takes the sphere onto itself while the open ball centred at 0 of radius r with its centre removed is taken to the complement of the closed ball with the same centre and radius.

r

Fig.7. Inversion i in the circle centred at 0 and of radius r in the plane R 2 defined by requiring lli(x)llllxll = r2 for x E R 2 \{0}.

Inversion is of course a bijective map of Rm\{O} onto itself and it follows from Examples 1.3.4 and 1.3.12 that it is continuous. Since it satisfies the condition ii= id (maps with such a property are called involutions) it is also a homeomorphism of the set Rm\{O} onto itself.

1.3.29. EXAMPLE. Stereographic projection. Let S be an m-dimensional sphere centred at c = (0,0, ... ,0,r) E Rm+l of radius r > 0. Then of course 0 ES. Let H be the hyperplane defined by the equation xm+l = 2r. Evidently H meets S in precisely the one point a= (0, 0, ... , 0, 2r).

Observe that every line joining the origin 0 to an arbitrary point x E S, where x = (x1,x2 , ,xm+l) =j:. 0 meets Hin exactly one point, which we denote by s(x). For, if the point s(x) = (1 - t)O + tx = tx is also to belong to H then txm+l = 2r. Thus t is determined uniquely by t = 2r/xm+l and so s(x) = (2r/xm+1)x for x E S\{0}. The map s: S\ {O} -+ H is called the stereographic projection of the sphere S from the pole 0 onto the hyperplane H. It is easy to see that s takes S\ { 0} onto H.

Recall from Example 1.3.28 that the inversion i in the sphere qmtred at 0 of radius 2r in the space Rm+l is given by the formula i(x) = (4r2 /llxll 2)x for x E Rm+l\{O}. If we also assume that x = (x1 , x2 , , xm+l) E S\ {O} then llx - cll 2 = r2 , that is llxll 2 -2xc+llcll 2 = r2 Since xc = xm+lr and llcll 2 = r2 we have llxll 2 = 2xm+lr so the


inversion i restricted to the sphere S is given by i(x) = (4r2 /2xm+lr)x = (2r /xm+l)x = s(x). Thus the stereographic projection s is the restriction of the inversion i to the sphere S. It follows therefore thats is a homeomorphism of the set S\{O} onto H. 1.3.30. COROLLARY. The m-dimensional sphere with one point removed and the m-dimensional Euclidean space Rm are homeomorphic.

a ,,,.----0----.........

,. ' ,. '

, '

-------

oc

0

Fig.8. The stereographic projection of the sphere S from the pole 0 onto the hyperplane H (Example 1.3.29).

The theory of uniform homeomorphism (respectively, homeomorphism} invariants, that is the properties of metric spaces which if enjoyed by one such space are enjoyed by all those which are uniformly homeomorphic (respectively, homeomorphic} to it is called the uniform topology (respectively, the topology) of metric spaces. The class of all metric spaces homeomorphic to a space X is called the topological type of the space X.

Fig.9. What may be obtained from the open disc by vario

engelking, sieklucki - topology a geometric approach

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