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Categorical Topology: Proceedings of the International Conference, Berlin, August 27th to September 2nd, 1978 PDF

432 Pages·1979·5.384 MB·English
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Lecture Notes ni Mathematics Edited yb .A Dold and .B Eckmann 719 lacirogetaC ygolopoT Proceedings of the International Conference, Berlin, August 27th to September 2nd, 1978 Edited yb .H Herrlich and .G BuerP galreV-regnirpS Berlin Heidelberg New kroY 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 yb photocopying machine or similar means, dna storage ni data banks. Under § 54 of the German Copyright waL where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to eb determined yb agreement with the wehsilbup © yb Springer-Verlag Berhn Heidelberg 9791 Printed ni Germany Printing dna binding: Beltz Offsetdruck, Hemsbach/Bergstr. 012345-0413/1412 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, Hajek and Mysior, Heldermann, Hoffmann, Husek, V 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): VI 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, Universitt Linz, A-~0%5 Linz-Auhof, Austria A. Frei 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 Universit~t Bremen, Fachsektion Mathematik, ~chterstrasse 33, D-25 Bremen, Fed. Rep. Germany R.-E. Hoffmann Universit~t Bremen, Fachsektion Mathematik, achterstrasse 33, D-28 Bremen, Fed. Rep. Germany M. Hu~ek Matematick~ ~stav, Karlovy University, Sokolovsk~ 83, 75600 Praha 8 - Karlin, Czechoslovakia P.T. Johnstone Pure Mathematics, qb Hill Lane, Cambridge, CB2 qSB, England K.H. Kamps Fachbereich Hathematik, Fermuniversit~t, Postfach 940, D-5800 Hagen, Fed. Rep. Germany .V Kannan Department of Mathematics, Hyderabad University, Wampally Station Road, Hyderabad - 500 OOq, India .H Kleisli Institut de MathSmatiques, Facu!t$ des Sciences, Universit@ de Fribourg, CH-1700 Fribourg, Suisse D° Leseberg Freie Universis~t Ber!in~ institut £~r Mathematik ,I H~ttenweg ,~ D-dO00 Berlin 33 S.E.J. Linton Department of Mathematics, Wesleyan University, Middletown, Connecticut 06457, USA R.B. L~schow Department of Mathematics and Computation, Faculty of Science, Universidad Tecnica del Estado, iZantiago, Chile J.L. MacDonald Department o£ Mathematics, University o£ British Columbia, Vancouver, B.C., Canada .T Marny Freie Universit~t Berlin, Institut f~r Mathematik ,I H~ttenweg ,9 D-~000 Berlin 33 E. Michael Department of Mathematics, University of Washington, Seattle, Washington 98~95, USA C.J. Mulvey 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 Universitt Bremen, Fachsektion Mathematik, Achterstra~e 3~, D-28 Bremen, Fed. Rep. Germany .G PreuB Freie Universitt 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 Matemticas 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- Zak~ad Matematyki, Wy~szej Szko~y Spirydonow 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 44 by J.B. Cooper and W. Schachermayer ........... 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 ?4 by G. Greve ................................... 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 by J.M. Harvey ................................ I02 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 ............... O~q ~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 .. 2o? 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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