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Compactness and the Axiom of Choice

  • Chapter
Categorical Topology

Abstract

In the absence of the axiom of choice four versions of compactness (A-, B-, C-, and D-compactness) are investigated.

  1. 1.

    C-compact spaces from the eprireflective hull in Haus of C-compact completely spaces.

  2. 2.

    Equivalent are:

  3. (a)

    the axiom of choice,

  4. (b)

    A-compact = D-compactness.

  5. (c)

    B-compactness = D-compactness,

  6. (d)

    C-compactness = D-compactness and complete regularity,

  7. (e)

    products of spaces with finite topologies are A-compact,>

  8. (f)

    products of A-compact spaces are A-compact ,

  9. (g)

    products of D-compact spaces are D-compact,

  10. (h)

    powers X k of 2-point discrete spaces are D-compact,

  11. (i)

    finite products of D-compact spaces are D-compact,

  12. (j)

    finite coproducts of D-compact spaces are D-compact,

  13. (k)

    D-compact Hausdorff spaces form an epireflective subcategory of Haus,

  14. (l)

    spaces with finite topologies are D-compact.

  15. 3.

    Equivalent are:

  16. (a)

    the Boolean prime ideal theorem,

  17. (b)

    A-compactness = B-compactness,

  18. (c)

    A-compactness and complete regularity = C-compactness,

  19. (d)

    products of spaces with finite undelying sets are A-compact,

  20. (e)

    products of A-compact Hausdorff spaces are A-compact,

  21. (f)

    powers X k of 2-point discrete spaces are A-compact,

  22. (g)

    A-compact Hausdorff spaces form an epireflective subcategory of Haus.

  23. 4.

    Equivalent are:

  24. (a)

    either the axiom of choice holds or every ultrafilter is fixed,

  25. (b)

    products of B-compact spaces are B-compact.

  26. 5.

    Equivalent are:

  27. (a)

    Dedekind-finite sets are finite,

  28. (b)

    every set carries some D-compact Hausdorff topology,

  29. (c)

    every T 1 has a T 1 — D-compactification,

  30. (d)

    Alexandroff-compactifications of discrete spaces are D-compact.

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Authors and Affiliations

  1. University of Bremen, Fachbereich 3, Postfach 33 04 40, 28334, Bremen, Germany

    Horst Herrlich

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  1. Horst Herrlich
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Editors and Affiliations

  1. Dipartimento di Matematica Pura ed Applicata, Università degli Studi de L’Aquila, Via Vetolo Loc Coppito, 67100, L’Aquita, Italy

    Eraldo Giuli

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Dedicated to My Friend Louis D. Nel on His Sixtieth Birthday

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© 1996 Kluwer Academic Publishers

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Herrlich, H. (1996). Compactness and the Axiom of Choice. In: Giuli, E. (eds) Categorical Topology. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0263-3_1

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  • DOI: https://doi.org/10.1007/978-94-009-0263-3_1

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-6602-0

  • Online ISBN: 978-94-009-0263-3

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