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Dimension Theory

A Selection of Theorems and Counterexamples

  • Book
  • © 2019

Overview

Authors:
  1. Michael G. Charalambous

    1. Department of Mathematics, University of the Aegean, Karlovassi, Greece

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  • Presupposed topological definitions and facts are carefully recalled
  • Includes numerous exercises proving presupposed facts
  • Includes numerous exercises of further interest
  • Helpful hints accompany exercises where necessary
  • Includes plenty of material that cannot be found elsewhere

Part of the book series: Atlantis Studies in Mathematics (ATLANTISSM, volume 7)

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Table of contents (32 chapters)

  1. Front Matter

    Pages i-x
    Download chapter PDF

  2. Topological Spaces

    • Michael G. Charalambous
    Pages 1-6

  3. The Three Main Dimension Functions

    • Michael G. Charalambous
    Pages 7-13

  4. The Countable Sum Theorem for Covering Dimension

    • Michael G. Charalambous
    Pages 15-22

  5. Urysohn Inequalities

    • Michael G. Charalambous
    Pages 23-26

  6. The Dimension of Euclidean Spaces

    • Michael G. Charalambous
    Pages 27-30

  7. Connected Components and Dimension

    • Michael G. Charalambous
    Pages 31-35

  8. Factorization and Compactification Theorems for Separable Metric Spaces

    • Michael G. Charalambous
    Pages 37-40

  9. Coincidence, Product and Decomposition Theorems for Separable Metric Spaces

    • Michael G. Charalambous
    Pages 41-44

  10. Universal Spaces for Separable Metric Spaces of Dimension at Most n

    • Michael G. Charalambous
    Pages 45-50

  11. Axiomatic Characterization of the Dimension of Separable Metric Spaces

    • Michael G. Charalambous
    Pages 51-60

  12. Cozero Sets and Covering Dimension dim0

    • Michael G. Charalambous
    Pages 61-74

  13. ψ-Spaces and the Failure of the Sum and Subset Theorems for dim 0

    • Michael G. Charalambous
    Pages 75-84

  14. The Inductive Dimension Ind 0

    • Michael G. Charalambous
    Pages 85-98

  15. Two Classical Examples

    • Michael G. Charalambous
    Pages 99-105

  16. The Gap Between the Covering and the Inductive Dimensions of Compact Hausdorff Spaces

    • Michael G. Charalambous
    Pages 107-114

  17. Inverse Limits and N-Compact Spaces

    • Michael G. Charalambous
    Pages 115-127

  18. Some Standard Results Concerning Metric Spaces

    • Michael G. Charalambous
    Pages 129-137

  19. The Mardešić Factorization Theorem and the Dimension of Metrizable Spaces

    • Michael G. Charalambous
    Pages 139-146

  20. A Metrizable Space with Unequal Inductive Dimensions

    • Michael G. Charalambous
    Pages 147-152
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Keywords

About this book

This book covers the fundamental results of the dimension theory of metrizable spaces, especially in the separable case. Its distinctive feature is the emphasis on the negative results for more general spaces, presenting a readable account of numerous counterexamples to well-known conjectures that have not been discussed in existing books. Moreover, it includes three new general methods for constructing spaces: Mrowka's psi-spaces, van Douwen's technique of assigning limit points to carefully selected sequences, and Fedorchuk's method of resolutions.


Accessible to readers familiar with the standard facts of general topology, the book is written in a reader-friendly style suitable for self-study. It contains enough material for one or more graduate courses in dimension theory and/or general topology. More than half of the contents do not appear in existing books, making it also a good reference for libraries and researchers.

Reviews

“The monograph contains a great deal of useful and up-to-date material on dimension theory; the exposition is transparent and well organized which makes it possible to use this work both as a textbook of dimension theory and a base of research projects in numerous areas.” (Vladimir Tkachuk, zbMATH 1471.54001, 2021)

Authors and Affiliations

  • Department of Mathematics, University of the Aegean, Karlovassi, Greece

    Michael G. Charalambous

Bibliographic Information

  • Book Title: Dimension Theory

  • Book Subtitle: A Selection of Theorems and Counterexamples

  • Authors: Michael G. Charalambous

  • Series Title: Atlantis Studies in Mathematics

  • DOI: https://doi.org/10.1007/978-3-030-22232-1

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer Nature Switzerland AG 2019

  • Hardcover ISBN: 978-3-030-22231-4Published: 18 October 2019

  • Softcover ISBN: 978-3-030-22234-5Published: 18 October 2020

  • eBook ISBN: 978-3-030-22232-1Published: 08 October 2019

  • Series ISSN: 1875-7634

  • Series E-ISSN: 2215-1885

  • Edition Number: 1

  • Number of Pages: X, 261

  • Number of Illustrations: 1 b/w illustrations

  • Topics: Topology, Analysis

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