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Lecture Notes in Mathematics
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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
719
Categorical Topology Proceedings of the International Conference, Berlin, August 27th to September 2nd, 1978
Edited by H. Herrlich and G. PreuB
Springer-Verlag Berlin Heidelberg New York 1979
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Editors Horst Herrlich Universit~t Bremen Fachbereich Mathematik AchterstraBe D-2800 Bremen
Gerhard Preu8 Freie Universita't Berlin Institut f~ir Mathematik I Hettenweg 9 D-1000 Berlin 33
AMS Subject Classifications (1970): 18-XX, 46 A 20, 46 E XX, 46 H XX, 46 MXX, 54-XX, 55DXX
ISBN 3-540-09503-9 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-38?-09503-9 Springer-Verlag NewYork Heidelberg Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publishew © by Springer-Verlag Berhn Heidelberg 1979 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
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PREFACE
The International Conference on Categorical Topology at the
Free University Berlin, from the 27th of August to the 2nd
of September 1978, brought together fifty mathematicians
from 16 countries of the world. The meeting was organized
by the editors of these proceedings. Especially, we would
like to thank Priv.-Doz. Dr. Thomas Marny (scientific
secretary) and Mrs. Karla Lautenschl~ger (congress bureau)
for their assistance. We are also grateful for the help of
the members of the Categorical Topology Research Group in
Berlin. We should not forget to thank the Deutsche Forschungs-
gemeinschaft, the Senator of Science and Research in Berlin
and the Free University Berlin for financial support.
And last not least we thank the Vice-President of the Free
University Berlin, Professor Heckelmann, for his willingness
to welcome the participants of the conference.
The scientific program of the conference consisted of talks
on various aspects of Categorical Topology. The subjects
treated in this volume may roughly be classified as follows:
I) Internal problems in particular topological categories
(Top, Unif, Conv, Near, Bitop):
Bentley and Herrlich, Brandenburg, BrHmmer, Frollk, V
Hajek and Mysior, Heldermann, Hoffmann, Husek, Kannan,
Kannan and Rajagopalan, Marny, Raghavan and Reilly,
Salbany, Schwarz, Wattel.
2) Topological categories
a) Function spaces and monoidal closed structures:
Greve , Nel, Porst and Wischnewsky, Wyler.
b) Connectedness-theories:
PreuS, Salicrup and V~zquez.
3) Topological functors and related concepts (structure
functors, (E,M)-functors, semi-topological functors,
initial completions):
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IV
B~rger, Harvey, Herrlich, Herrlich and Strecker,
Nakagawa, Tholen, Wischnewsky.
4) Applications
a) topological algebra:
Smith-Thomas
b) algebraic topology:
Frei and Kleisli, Hardie, Heath and Kamps, Mac
Donald
c) functional analysis:
Banaschewski, Cooper and Schachermayer, Mulvey,
Nel, Semadeni and Zidenberg-Spirydonow
d) statistic metric spaces
LHschow
5) Topoi:
Johnstone, Linton and Par~.
Thus, the papers give a survey on the recent state of re-
search in Categorical Topology and its applications. Finally,
we would like to express our thanks to the Springer Verlag
for publishing the proceedings of this conference in the Lec-
ture Notes series.
H. Herrlich G. PreuB
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Address list of authors and speakers
B. Banaschewski Department of Mathematics, McMaster University, Hamilton, Ontario L8S %K~, Canada
H.L. Bentley Department of Mathematics, University of Toledo, Toledo, Ohio 43606, USA
R. BSrger Fachbereich Mathematik, Fernuniversit~t, Postfach 9%0, D-5800 Hagen, Fed. Rep. Germany
H. Brandenburg Freie Universit~t Berlin, Insti~ut fur Mathematik I, H~ttenweg 9, D-tO00 Berlin 33
G.C.L. Br~mmer Department of Mathematics, University of Cape Town, Private Bag, Rondebosch 9700, Republic of South Africa
J.B. Cooper institut £Hr Mathematik, Universit[t Linz, A-~0%5 Linz-Auhof, Austria
A. F r e i Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada
5. Frolik Matematick@ ~stav ~SAV, Zitn[ 25, 775 69 Praha 7, Czechoslovakia
G. Greve Fachbereich Mathematik, Fernuniversit~t, L~tzowstr. 725, D-5800 Hagen, Fed. Rep. Germany
D.W. Haj ek Department of Mathematics, University of Puerto Rico, Mayaguez, Puerto Rico 00708
K.A. Hardie Department o£ Mathematics, University of Cape Town, Rondebosch 7900, Republic of South Africa
J.M. Harvey Department of Mathematics, University of Rhodesia, P.0.Box MP 169, Salisbury, Rhodesia
P.R. Heath Department of Mathematics, Memorial University or Newfoundland, St. John's, Newfoundland, Canada, ATB 3X7
N.C. Heldermann Zentralblatt fHr Mathematik, 0tto-Suhr- Allee 26, D-7000 Berlin 70
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Vl
H. Herrlich
R.-E. Hoffmann
M. Hu~ek
P.T. Johnstone
K.H. Kamps
V. Kannan
H. Kleisli
D° Leseberg
S.E.J. Linton
R.B. L~schow
J.L. MacDonald
T. Marny
E. Michael
C.J. Mulvey
Universit~t Bremen, Fachsektion Mathematik, ~chterstrasse 33, D-25 Bremen, Fed. Rep. Germany
Universit~t Bremen, Fachsektion Mathematik, achterstrasse 33, D-28 Bremen, Fed. Rep. Germany
Matematick~ ~stav, Karlovy University, Sokolovsk~ 83, 75600 Praha 8 - Karlin, Czechoslovakia
Pure Mathematics, qb Hill Lane, Cambridge, CB2 qSB, England
Fachbereich Hathematik, Fermuniversit~t, Postfach 940, D-5800 Hagen, Fed. Rep. Germany
Department of Mathematics, Hyderabad University, Wampally Station Road, Hyderabad - 500 OOq, India
Institut de MathSmatiques, Facu!t$ des Sciences, Universit@ de Fribourg, CH-1700 Fribourg, Suisse
Freie Universis~t Ber!in~ institut £~r Mathematik I, H~ttenweg ~, D-dO00 Berlin 33
Department of Mathematics, Wesleyan University, Middletown, Connecticut 06457, USA
Department of Mathematics and Computation, Faculty of Science, Universidad Tecnica del Estado, iZantiago, Chile
Department o£ Mathematics, University o£ British Columbia, Vancouver, B.C., Canada
Freie Universit~t Berlin, Institut f~r Mathematik I, H~ttenweg 9, D-~000 Berlin 33
Department of Mathematics, University of Washington, Seattle, Washington 98~95, USA
Mathematics Division, University of Sussex, Falmer, Brighton, BN~ 9QH, England
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Vll
A. Mysior Institute of Mathematics, University of Gda~sk, 80-952 Gda~sk, Wita Stwosza 57, Poland
R. Nakagawa Department of Mathematics, University of Tsukuba, !baraki, Japan
L.D. Nel Department of Mathematics, Carieton University, Ottawa, Ontario, Canada KI S 5B6
R. Par$ Mathematics Department, Dalhousie Univer- sity, Halifax, New Scotland, Canada B3H ~H8
H.-E. Pomst Universit[t Bremen, Fachsektion Mathematik, Achterstra~e 3~, D-28 Bremen, Fed. Rep. Germany
G. PreuB Freie Universit[t Berlin, Institut fur Mathematik I, H~ttenweg 9, D-1000 Berlin 33
H. Pust Freie Universit~t Berlin, Institut fur Mathematik I, H~ttenweg 9, D-IOOO Berlin 33
T.G. Raghavan Department of Mathematics, University of Auckland, Private Bag, Auckland, New Zealand
M. Rajagopalan Universidad de los Andes, Facultad de Ciencas, Departemento de Matematicas, Merida, Venezuela
I.L. Reilly Department o£ Mathematics, University of Auckland, Private Bag, Auckland, New Zealand
M.D. Rice Department of Mathematics, George Mason University, Fairfax, Virginia 22030, USA
S. Salbany Department of Mathematics, University of Cape Town, Rondebosch 7700, Republic of South Africa
G. Salicrup Instituto de Matem&ticas U.N.A.M., Ciudad Universitaria, M6xico 20, D.F.
W. Schachermayer Institut fur Mathematik, Universit~t Linz, A-~045 Linz-Auhof, Austria
F. Schwarz Institut fur Mathematik, Technische Univer- sit,t, Welfengarten q, D-3000 Hannover I, Fed. Rep° Germany
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Vfll
Z. Semadeni Instytut Matematyczny, Polskiej Akademii Nauk, ul. Sniadeckich 8, skr. poczt. ~37, 00-9~0 Warszawa, Poland
B.V. Smith Thomas Department of Mathematical Sciences, Memphis State University, Memphis, Tennessee 38q~2, USA
G.E. Strecker Department of Mathematics, Kansas State University, Manhattan, Kansas 66502, USA
W. Tholen Pachbereich Mathematik, Fernuniversitgt, Postfach 940, D-~800 Hagen, Fed. Rep. Germany
R. V&zquez Insti~uto de Matem[ticas U.N.A.M., Ciudad Universitaria, M6xico 20, D.F.
E. Wattel Subfaculteit Wiskunde, Wiskundig Seminarium, Vrije Universiteit, ~e Boelelaan ~081, Amsterdam - ~0~, The Netherlands
R. Wiegandt Mathematical Instituse of the Hungarian Academy o£ 3ciences, H-I053 Budapest, Re&itanoda u. i~-iS, Hungary
M. Wischnewsky Pachbereich Mathematik, Universit~t Bremen, Kufsteiner StraBe, D-2880 Bremen 33, Fed. Rep. Germany
O. Wyler Department of Mathematics, Carnegie-Mellon University, Pittsburgh, PA ~52~3, USA
H. Zidenberg- Spirydonow
Zak~ad Matematyki, Wy~szej Szko~y Pedagogicznej, ul. Wielkopolska ~5, 70-387 Jzczecin, Poland
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CONTENTS
Recovering a space from its Banach sheaves
by B. Banaschewski ............................
Completeness is productive
by H.L. Bentley and H. Herrlich ............... 13
Legitimacy of certain topological categories
by R. B6rger .................................. 78
0n ~-normal spaces
by H. Brandenburg 24
Two procedures in bitopology
by G.C.L. Br~mmer ............................. 35
Saks spaces and vector valued measures
by J.B. Cooper and W. Schachermayer ........... 44
A questmon in categorical shape theory: When is a shape-invariant functor a Man extension?
by ~. Frei and H. Kleisli .....................
The finest functor preserving the Baire sets
by Z. Frollk .................................. 63
Lifting closed and monoidal structures along semitopological functors
by G. Greve ................................... ?4
On non-simplicity of topological categories
by D.W. Hajek and A. Mysior ................... 84
Kan lift-extensions in C.G. Haus~
by K.A. Hardie ................................ 94
Topological functors from factorization
I02 by J.M. Harvey ................................
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Groupoids and classification sequences
by P.R. Heath and K.H. Kamps ................... qq2
Concentrated nearness spaces
by N.C. Heldermann ............................. q22
Initial and final completions
by H. Herrlich ................................. 137
Algebra ~ Topology
by H. Herrlica and G.£. Strecker ............... q~O
~opological spaces admibting a "dual"
by R.-E. Hoffmann .............................. 157
Special classes of compact spaces
by H. Hu~ek .................................... 167
Pairs of topologies with same family of continuous self-maps
by V. Kannan ................................... 176
Hereditarily locally compact separable spaces
by V. Kannan and M. Rajagopalan ................ fiS~
Injectives in topoi, I: Representing coalgebras as algebras
by 2.E.J. Linton and R. ~arS ................... q96
InVectives in topoi, II: Connections with the axzom of choice
by P.T. Johnstone, F.E.J. Linton, and R. Pars . . 2 o ?
Categories of statistic-metric spaces (a co-universal construction)
by R.B. L~schow ................................ 2fl7
A categorical approach to primary and secondary operations in topology
by J.L. MacDonald .............................. 225
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XI
Limit-metrizability of limit spaces and uniform limit spaces
by T. Marny ..................................... 234
Banach spaces over a compact space
by C.J. Mulvey .................................. 243
A note on (S,M)-runctors
by R. Nakagawa .................................. 250
Convenient topological algebra and reflexive objects
by L,D. Nel ..................................... 259
Existence and applications of monoidally closed structures in topological categories
by H.-E. Porst and M.B. Wischnewsky ............. 277
Connection properties in topological categories and related topics
by G. Preu~ ..................................... 293
On projective and injective objects in some topological categories
by T.G. Raghavan and I.L. Reilly ................ 308
An embedding characterization o~ compact spaces
by S. Salbany ................................... 3~6
Connection and disconnection
by G. Salicrup and R. V~zquez ................... 326
Connections between convergence and nearness
by F. Schwarz ................................... 34~
Functors on categories of ordered topological spaces
by Z. Semadeni and H. Zidenberg-Spirydonow ...... 358
On the coproduct o£ the topological groups ~ and
by B.V. Smith Thomas ............................ 37~
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XII
LiZting semi£inal li£tings
by W. Tholen ....................................
Normally supercompact spaces and convexity preservzng maps
by E. War,el ....................................
Structure £unctors - Representation and existence theorems -
by M.B. Wischnewsky .............................
Function spaces in topological categories
by O. Wyler .....................................
3?6
386
395
~d q
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