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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

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

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):

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

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

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

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

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

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 ................................

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

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~

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