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General Topology II

Compactness, Homologies of General Spaces

  • Book
  • © 1996

Overview

Editors:
  1. A. V. Arhangel’skii

    1. Department of Mathematics, University of Moscow, Moscow, Russia
      Department of Mathematics, Ohio University, Athens, USA

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Part of the book series: Encyclopaedia of Mathematical Sciences (EMS, volume 50)

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-3
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  2. Introduction

    • A. V. Arhangel’skii
    Pages 4-4

  3. Compactness and Its Different Forms: Separation Axioms

    • A. V. Arhangel’skii
    Pages 5-13

  4. Compactness and Products

    • A. V. Arhangel’skii
    Pages 13-19

  5. Continuous Mappings of Compact Spaces

    • A. V. Arhangel’skii
    Pages 19-28

  6. Metrizability Conditions for Compact, Countably Compact and Pseudocompact Spaces

    • A. V. Arhangel’skii
    Pages 28-34

  7. Cardinal Invariants in the Class of Compacta

    • A. V. Arhangel’skii
    Pages 34-59

  8. Compact Extensions

    • A. V. Arhangel’skii
    Pages 59-76

  9. Compactness and Spaces of Functions

    • A. V. Arhangel’skii
    Pages 76-93

  10. Algebraic Structures and Compactness. A Review of the Most Important Results

    • A. V. Arhangel’skii
    Pages 93-108

  11. Back Matter

    Pages 108-258
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Keywords

About this book

Compactness is related to a number of fundamental concepts of mathemat­ ics. Particularly important are compact Hausdorff spaces or compacta. Com­ pactness appeared in mathematics for the first time as one of the main topo­ logical properties of an interval, a square, a sphere and any closed, bounded subset of a finite dimensional Euclidean space. Once it was realized that pre­ cisely this property was responsible for a series of fundamental facts related to those sets such as boundedness and uniform continuity of continuous func­ tions defined on them, compactness was given an abstract definition in the language of general topology reaching far beyond the class of metric spaces. This immensely extended the realm of application of this concept (including in particular, function spaces of quite general nature). The fact, that general topology provided an adequate language for a description of the concept of compactness and secured a natural medium for its harmonious development is a major credit to this area of mathematics. The final formulation of a general definition of compactness and the creation of the foundations of the theory of compact topological spaces are due to P.S. Aleksandrov and Urysohn (see Aleksandrov and Urysohn (1971)).

Editors and Affiliations

  • Department of Mathematics, University of Moscow, Moscow, Russia

    A. V. Arhangel’skii

  • Department of Mathematics, Ohio University, Athens, USA

    A. V. Arhangel’skii

Bibliographic Information

  • Book Title: General Topology II

  • Book Subtitle: Compactness, Homologies of General Spaces

  • Editors: A. V. Arhangel’skii

  • Series Title: Encyclopaedia of Mathematical Sciences

  • DOI: https://doi.org/10.1007/978-3-642-77030-2

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 1996

  • Softcover ISBN: 978-3-642-77032-6Published: 16 December 2011

  • eBook ISBN: 978-3-642-77030-2Published: 06 December 2012

  • Series ISSN: 0938-0396

  • Edition Number: 1

  • Number of Pages: VII, 256

  • Additional Information: Original Russian edition published by VINITI Moscow, 1989

  • Topics: Algebraic Topology, Topological Groups, Lie Groups, K-Theory

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