Encyclopaedia of Mathematical Sciences

Transcription

1 Encyclopaedia of Mathematical Sciences Volume 50 Editor-in-Chief: R. V. Gamkrelidze

2 Springer Berlin Heidelberg New ork Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo

3 A. V. Arhangel'skii (Ed.) General Topology II Compactness, Homologies of General Spaces Springer

4 Consulting Editors of the Series: A.A. Agrachev, A.A. Gonchar, E.F. Mishchenko, N.M. Ostianu, V.P. Sakharova, A.B. Zhishchenko Title of the Russian edition: Itogi nauki i tekhniki, Sovremennye problemy matematiki, Fundamental'nye napravleniya, Vol. 50 Obshchaya Topologiya-2 Publisher VINITI, Moscow 1989 Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme General topology. - Berlin; Heidelberg; New York; London; Paris; Tokyo; Hong Kong; Barcelona: Springer. Einheitssacht.: Obseaja topologija <engl. > Li teraturangaben NE: EST 2. Compactness, homologies of general spaces / A. V. Arhangel'skii (ed.) (Encyclopaedia of mathematical sciences; Vol. SO) ISBN-13: NE: Archangel 'skij, Aleksandr V.; GT Mathematics Subject Classification (1991):54DXX, 55NXX ISBN-13: DOl: / e-isbn-13: This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned. specifically the rights of translation. reprinting. reuse of illustrations. recitation, broadcasting. reproduction on microfilm or in any other way. and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September in its current version. and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 1996 Softcover reprint of the hardcover 1st edition 1996 Typesetting: Camera-ready copy from the translator using a Springer TEX macro package SPIN: Printed on acid:tree paper

5 List of Editors, Authors and Translators Editor-in-Chief R. V. Gamkrelidze, Russian Academy of Sciences, Steklov Mathematical Institute, ul. Vavilova 42, Moscow, Institute for Scientific Information (VINITI), ul. Usievicha 20a, Moscow, Russia, Consulting Editor A. V. Arhangel' skii, Department of Mathematics, University of Moscow, Leninskie Gory, Moscow, Russia, and Department of Mathematics, Ohio University, 321 Morton Hall, Athens, Ohio , USA, Authors A. V.Arhangel'skii, Department of Mathematics, University of Moscow, Leninskie Gory, Moscow, Russia, and Department of Mathematics, Ohio University, 321 Morton Hall, Athens, Ohio , USA, E. G. Sklyarenko, Department of Mathematics, University of Moscow, Leninskie Gory, Moscow, Russia Translator J. M. Lysko, College of Arts and Science, Science Division, Widener University, One University Place, Chester, PA 19013, USA,

6 Contents I. Compactness A. V. Arhangel'skii 1 II. Homology and Cohomology Theories of General Spaces E. G. Sldyarenko 119 Author Index 247 SUbject Index 251

7 I. Compactness A.V. Arhangel'skii Translated from the Russian by Janusz M. Lysko Contents Introduction Compactness and Its Different Forms: Separation Axioms Different Definitions of Compactness Relative Compactness Countable Compactness Relative Countable Compactness Pseudo compact Spaces Separation Axioms and Properties Related to Compactness Star Characterizations of Countable Compactness and Pseudo compactness Compactness and Products Tikhonov's Theorem on Compactness of the Product Products of Countably Compact Spaces Products of Pseudo compact Spaces Total Countable Compactness and Total Pseudo compactness Compactness with Respect to a Fixed Ultrafilter ( ~ - C o m p a c... t n e s s ) :-Products of Compact Spaces Continuous Mappings of Compact Spaces Theorem on Compactness of the Image and Its Consequences Continuous Images of "Standard" Compacta Open Mappings of Compacta and Dimension... 21

8 2 A.V. Arhangel'skii 3.4. MardesiC's Factorization Theorem Continuous Images of Ordered Compacta Pseudocompactness and Continuous Mappings Continuous Mappings and Extremally Disconnected Compacta Scattered Compacta and Their Images Metrizability Conditions for Compact, Countably Compact and Pseudocompact Spaces Classical Results and the Theorem of Chaber Theorems of Dow and Tkachenko Point-countable and cr-point-finite Bases Quasi-developments and 58-Bases Strongly No-Noetherian Bases Rank of a Base and Metrizability Conditions for Compacta Symmetrics and Metrizability of Compacta Cardinal Invariants in the Class of Compacta Network Weight, Diagonal Number and Weight of Compacta Pseudo character and Character in the Class of Compacta First Countable Compacta Perfectly Normal Compacta Continuous Images of First Countable Compacta Sequential Compacta and the First Axiom of Count ability Almost Everywhere Corson Compact Spaces and No-Monolithicity Compacta of Countable Tightness Mappings of Compacta onto Tikhonov Cubes F Dyadic Compacta Supercompacta and Extensions of the Class of Dyadic Compacta Compact Extensions General Remarks about Compact Extensions Compact TI-Extensions Embedding Topological Spaces into Compact TI-Spaces of Countable Weight Compact Hausdorff Extensions, Relation of Subordination Compact Extensions of Locally Compact Hausdorff Spaces Duality Between Properties of a Space and of Its Remainder Compact Extensions and Cardinal Invariants Compact Hausdorff Extensions and Perfect Mappings Properties of the Cech-Stone Extension Closing Remarks Concerning Compact Hausdorff Extensions... 75

9 I. Compactness 3 7. Compactness and Spaces of Functions Natural Topologies on Spaces of Functions Joint Continuity and Compact-Open Topology Stone-Weierstrass Theorem Convex Compact Sets and Krein-Milman Theorem Theorem of Alaoglu and Convex Hulls of Compacta Fixed-Point Theorems for Continuous Mappings of Convex Compacta Milyutin Compact Spaces Dugundji Compact Spaces Algebraic Structures and Compactness - A Review of the Most Important Results Compacta and Ideals in Rings of Functions Spectrum of a Ring. Zariski Topology The Space of Maximal Ideals of a Commutative Banach Algebra The Stone Space of a Boolean Algebra Pontryagin's Duality Theory Compact Extensions of Topological Groups. Almost Periodic Functions Compacta and Namioka's Theorem About Joint Continuity of Separately Continuous Functions Fragmentable and Strongly Fragmentable Compacta and Radon-Nikod:0n Compact Spaces Hilbert Modules over C* -Algebras of Continuous Functions on Compacta Compact Subsets of Topological Fields Locally Compact Topological Groups and Paracompactness Final Remarks References