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Lecture Notes in Mathematics For information about Vols. 1-489, please contact your bookseller or Springer-Verlag. Vol. 521: G. Cherlin, Model Theoretic Algebra - Selected Topics. IV, 234 pages. 1976.

1 Lecture Notes in Mathematics For information about Vols , please contact your bookseller or Springer-Verlag. Vol. 521: G. Cherlin, Model Theoretic Algebra - Selected Topics. IV, 234 pages Vol. 490: The Geometry of Metric and Linear Spaces. Proceedings Edited by L. M. Kelly. X, 244 pages Vol. 522: C. 0. Bloom and N. D. Kazarinoff, Short Wave Radiation Problems in Inhomogeneous Media: Asymptotic Solutions. V. 104 pages Vol. 491: K. A. Broughan, Invariants for Real-Generated Uniform Topological and Algebraic Categories. X, 197 pages Vol. 492: lnfinitary Logic: In Memoriam Carol Karp. Edited by D. W. Kueker. VI, 206 pages Vol. 493: F. W. Kamber and P. Tondeur, Foliated Bundles and Characteristic Classes. XIII, 208 pages Vol. 494: A Cornea and G. Licea. Order and Potential Resolvent Families of Kernels. IV, 154 pages Vol. 495: A. Kerber, Representations of Permutation Groups II. V, 175 pages Vol. 496: L. H. Hodgkin and V. P. Snaith, Topics ink-theory. Two Independent Contributions. Ill, 294 pages Vol. 497: Analyse Harmonique sur les Groupes de Lie. Proceedings Edite par P. Eymard et al. VI, 710 pages Vol. 498: Model Theory and Algebra. A Memorial Tribute to Abraham Robinson. Edited by D. H. Saracino andv. B. Weispfenning. X, 463 pages Vol. 499: Logic Conference, Kiel Proceedings. Edited by G. H. Muller, A. Oberschelp, and K. Potthoff. V, 651 pages Vol. 500: ProofTheory Symposion, Kiel1974. Proceedings. Edited by J. Diller and G. H. Muller. VIII, 383 pages Vol. 501: Spline Functions, Karlsruhe Proceedings. Edited by K. B<lhmer, G. Meinardus, and W. Schempp. VI, 421 pages Vol. 502: Janos Galambos, Representations of Real Numbers by Infinite Series. VI, 146 pages Vol. 503: Applications of Methods of Functional Analysis to Problems in Mechanics. Proceedings Edited by P. Germain and B. Nayroles. XIX, 531 pages Vol. 504: S. Lang and H. F. Trotter, Frobenius Distributions m GL2 -Extensions. Ill, 274 pages Vol. 505: Advances in Complex Function Theory. Proceedings 1973/74. Edited by W. E. Kirwan and L. Zalcman. VIII, 203 pages Vol. 506: Numerical Analysis, Dundee Proceedings. Edited by G. A. Watson. X, 201 pages Vol. 507: M. C. Reed, Abstract Non-Lmear Wave Equations. VI, 128 pages ' Vol. 508: E. Seneta, Regularly Varying Functions. V, 112 pages Vol. 523: S. A. Albeverio and R. J. H0egh-Krohn, Mathematical Theory of Feynman Path Integrals. IV, 139 pages Vol. 524: Seminaire Pierre Lelong (Analyse) Annee 1974/75. Edite par P. Lelong. V, 222 pages Vol. 525: Structural Stability, the Theory of Catastrophes, and Applications in the Sciences. Proceedings Edited by P. Hilton. VI, 408 pages Vol. 526: Probability in Banach Spaces. Proceedings Edited by A. Beck. VI, 290 pages Vol. 527: M. Denker, Ch. Grillenberger, and K. Sigmund, Ergodic Theory on Compact Spaces. IV, 360 pages Vol. 528: J. E. Humphreys, Ordinary and Modular Representations of Chevalley Groups. Ill, 127 pages. 1g76. Vol. 529: J. Grandell, Doubly Stochastic Poisson Processes. X, 234 pages. 1g76. Vol. 530: S. S. Gelbart, Weil's Representation and the Spectrum of the Metaplectic Group. VII, 140 pages Vol. 531: Y.-C. Wong, The Topology of Uniform Convergence on Order-Bounded Sets. VI, 163 pages Vol. 532: Theorie Ergodique. Proceedings Edite par J.-P. Conze and M.S. Keane. VIII, 227 pages Vol. 533: F. R. Cohen, T. J. Lada, and J. P. May, The Homology of Iterated Loop Spaces. IX, 490 pages Vol. 534: C. Preston, Random Fields. V, 200 pages Vol. 535: Singularites d'applications Differentiables. Plans-sur-Sex Edite par 0. Burlet et F. Ronga. V, 253 pages Vol. 536: W. M. Schmidt, Equations over Finite Fields. An Elementary Approach. IX, 267 pages Vol. 537: Set Theory and Hierarchy Theory. Bierutowice, Poland A Memorial Tribute to Andrzej Maslowski. Edited by W. Marek, M. Srebrny and A. Zarach. XIII, 345 pages Vol. 538: G. Fischer, Complex Analytic Geometry. VII, 201 pages Vol. 539: A. Badrikian, J. F. C. Kingman et J. Kuelbs, Ecole d'ete de Probabilites de Saint Flour V Edite par P.-l. Hennequin. IX, 314 pages Vol. 540: Categorical Topology, Proceedings Edited by E. Binz and H. Herrlich. XV, 719 pages V.ol. 509: D. E. Blair, Contact Manifolds in Riemannian Geometry. VI, 146 pages Vol. 541: Measure Theory, Oberwolfach Proceedings. Edited by A. Bellow and D. K<llzow. XIV, 430 pages Vol. 510: V. Poenaru, Singularites C"' en Presence de Symetrie. V, 174 pages Vol. 542: D. A. Edwards and H. M. Hastings, Cech and Steenrod Homotopy Theories with Applications to Geometric To~logy. VII, 296 pages Vol. 511: Seminaire de Probabilites X. Proceedings 1g74/75. Edite par P. A. Meyer. VI, 593 pages Vol. 512: Spaces of Analytic Functions, Kristiansand, Norway Proceedings. Edited by 0. B. Bekken, B. K. 0ksendal, and A. Stray. VIII, 204 pages Vol. 513: R. B. Warfield, Jr. Nilpotent Groups. VIII, 115 pages Vol. 514: Seminaire Bourbaki vol. 1974n5. Exposes V, 276 pages Vol. 515: B~cklund Transformations. Nashville, Tennessee Proceedings. Edited by R. M. Miura. VIII, 295 pages Vol. 516: M. L. Silverstein, Boundary Theory for Symmetric Markov Processes. XVI, 314 pages Vol. 517: S. Glasner, Proximal Flows. VIII, 153 pages Vol. 518: Seminaire de Theorie du Potential, Proceedings Paris Edite par F. Hirsch et G. Mokobodzki. VI, 275 pages Vol. 519: J. Schmets, Espaces de Fonctions Continues. XII, 150 pages Vol. 520: R. H. Farrell, Techniques of Multivariate Calculation. X, 337 pages Vol. 543: Nonlinear Operators and the Calculus of Variations, Bruxelles Edited by J. P. Gossez, E. J. Lami Dozo, J. Mawhin, and L. Waelbroeck, VII, 237 pages Vol. 544: Robert P. Langlands, On the Functional Equations Satisfied by Eisenstein Series. VII, 337 pages Vot. 545: Noncommutative Ring Theory. Kent State Edited by J. H. Cozzens and F. L. Sandomierski. V, 212 pages Vol. 546: K. Mahler, Lectures on Transcendental Numbers. Edited and Completed by B. Divis and W. J. Le Veque. XXI, 254 pages Vol. 547: A. Mukherjea and N. A. Tserpes, Measures on Topological Semigroups: Convolution Products and Random Walks. V, 197 pages Vol. 548: D. A. Hejhal, The Selberg Trace Formula for PSL (2,1R). Volume I. VI, 516 pages Vol. 549: Brauer Groups, Evanston Proceedings. Edited by D. Zelinsky. V, 187 pages Vol. 550: Proceedings of the Third Japan - USSR Symposium on Probability Theory. Edited by G. Maruyama and J. V. Prokhorov. VI, 722 pages continuation on page 423

2 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

3 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 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-38? 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 Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/

4 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 Forschungsgemeinschaft, 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):

5 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 research 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 Lecture Notes series. H. Herrlich G. PreuB

6 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. H. Brandenburg Freie Universit~t Berlin, Insti~ut fur Mathematik I, H~ttenweg 9, D-tO00 Berlin 33 G.C.L. Br~mmer J.B. Cooper Department of Mathematics, University of Cape Town, Private Bag, Rondebosch 9700, Republic of South Africa institut Hr Mathematik, Universit[t Linz, A-~0%5 Linz-Auhof, Austria A. Frei Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada 5. Frolik ~stav ~SAV, Zitn[ 25, Praha 7, Czechoslovakia G. Greve Fachbereich Mathematik, Fernuniversit~t, L~tzowstr. 725, D-5800 Hagen, Fed. Rep. D.W. Haj ek K.A. Hardie J.M. Harvey P.R. Heath N.C. Heldermann Department of Mathematics, University of Puerto Rico, Mayaguez, Puerto Rico Department o Mathematics, University of Cape Town, Rondebosch 7900, Republic of South Africa Department of Mathematics, University of Rhodesia, P.0.Box MP 169, Salisbury, Rhodesia Department of Mathematics, Memorial University or Newfoundland, St. John's, Newfoundland, Canada, ATB 3X7 Zentralblatt fhr Mathematik, 0tto-Suhr- Allee 26, D-7000 Berlin 70

7 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. Universit~t Bremen, Fachsektion Mathematik, achterstrasse 33, D-28 Bremen, Fed. Rep. Matematick~ ~stav, Karlovy University, Sokolovsk~ 83, Praha 8 - Karlin, Czechoslovakia Pure Mathematics, qb Hill Lane, Cambridge, CB2 qsb, England Fachbereich Hathematik, Fermuniversit~t, Postfach 940, D-5800 Hagen, Fed. Rep. Department of Mathematics, Hyderabad University, Wampally Station Road, Hyderabad 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

8 Vll A. Mysior Institute of Mathematics, University of Gda~sk, 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 University, Halifax, New Scotland, Canada B3H ~H8 H.-E. Pomst Universit[t Bremen, Fachsektion Mathematik, Achterstra~e 3~, D-28 Bremen, Fed. Rep. 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 M.D. Rice Department o Mathematics, University of Auckland, Private Bag, Auckland, New Zealand 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 Universit,t, Welfengarten q, D-3000 Hannover I, Fed. Rep

9 Vfll Z. Semadeni Instytut Matematyczny, Polskiej Akademii Nauk, ul. Sniadeckich 8, skr. poczt. ~37, 00-9~0 Warszawa, Poland B.V. Smith Thomas G.E. Strecker Department of Mathematical Sciences, Memphis State University, Memphis, Tennessee 38q~2, USA Department of Mathematics, Kansas State University, Manhattan, Kansas 66502, USA W. Tholen Pachbereich Mathematik, Fernuniversitgt, Postfach 940, D-~800 Hagen, Fed. Rep. 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. 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, Jzczecin, Poland

10 CONTENTS Recovering a space from its Banach sheaves by B. Banaschewski... Completeness is productive by H.L. Bentley and H. Herrlich Legitimacy of certain topological categories by R. B6rger n ~-normal spaces by H. Brandenburg 24 Two procedures in bitopology by G.C.L. Br~mmer Saks spaces and vector valued measures 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 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 Kan lift-extensions in C.G. Haus~ by K.A. Hardie Topological functors from factorization by J.M. Harvey... I02

11 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 Algebra ~ Topology by H. Herrlica and G.. Strecker... q~o ~opological spaces admibting a "dual" by R.-E. Hoffmann Special classes of compact spaces by H. Hu~ek Pairs of topologies with same family of continuous self-maps by V. Kannan 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

12 XI Limit-metrizability of limit spaces and uniform limit spaces by T. Marny Banach spaces over a compact space by C.J. Mulvey A note on (S,M)-runctors by R. Nakagawa Convenient topological algebra and reflexive objects by L,D. Nel Existence and applications of monoidally closed structures in topological categories by H.-E. Porst and M.B. Wischnewsky Connection properties in topological categories and related topics by G. Preu~ On projective and injective objects in some topological categories by T.G. Raghavan and I.L. Reilly An embedding characterization o~ compact spaces by S. Salbany... 3~6 Connection and disconnection by G. Salicrup and R. V~zquez Connections between convergence and nearness by F. Schwarz... 34~ Functors on categories of ordered topological spaces by Z. Semadeni and H. Zidenberg-Spirydonow On the coproduct o the topological groups ~ and by B.V. Smith Thomas... 37~

13 XII LiZting semi inal li tings by W. Tholen... 3?6 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... ~d q

Lecture Notes 1n Mathematics

Lecture Notes 1n Mathematics For information about Vols. 1-489, please contact your bookseller or Springer-Verlag. Vol. 490: The Geometry of Metric and Linear Spaces. Proceedings 1974. Edited by L. M.