Skip to main content
SpringerLink
Log in

Compactness and the axiom of choice

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

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

Typical results:

  1. 1.

    C-compact spaces form the epireflective hull in Haus of A-compact completely regular spaces.

  2. 2.

    Equivalent are:

  3. (a)

    the axiom of choice,

  4. (b)

    A-compactness = 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.

  1. 3.

    Equivalent are:

  2. (a)

    the Boolean prime ideal theorem,

  3. (b)

    A-compactness = B-compactness,

  4. (c)

    A-compactness and complete regularity = C-compactness,

  5. (d)

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

  6. (e)

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

  7. (f)

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

  8. (g)

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

  1. 4.

    Equivalent are:

  2. (a)

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

  3. (b)

    products of B-compact spaces are B-compact.

  1. 5.

    Equivalent are:

  2. (a)

    Dedekind-finite sets are finite,

  3. (b)

    every set carries some D-compact Hausdorff topology,

  4. (c)

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

  5. (d)

    Alexandroff-compactifications of discrete spaces and D-compact.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Alas, O. T.: The axiom of choice and two particular forms of Tychonoff theorem, Portugal. Math. 28 (1969), 75–76.

    Google Scholar 

  • Alexandroff, P. and Urysohn, P.: Mémoire sur les espaces topologiques compacts, Verh. Nederl. Akad. Wetensch. Aft. Naturk. Sect. 1 14 (1929), 1–96.

    Google Scholar 

  • Banaschewski, B.: Compactification and the axiom of choice, Unpublished manuscript, 1979.

  • Bentley, H. L. and Herrlich, H.: Compactness and rings of continuous functions — without the axiom of choice, To appear (1995+).

  • Blass, A.: A model without ulrafilters, Bull. Acad. Sci. Polon., Sér. Sci. Math. Astr. Phys. 25 (1977), 329–331.

    Google Scholar 

  • Čech, E.: On bicompact spaces, Ann. Math. 38 (1937), 823–844.

    Google Scholar 

  • Chandler, R. E.: An alternative construction of βX and νX, Proc. Amer. Math. Soc. 32 (1972), 315–318.

    Google Scholar 

  • Comfort, W. W.: A theorem of Stone-Čech type, and a theorem of Tychonoff type, without the axiom of choice; and their realcompact analogues, Fund. Math. 63 (1968), 97–110.

    Google Scholar 

  • Feferman, S.: Some applications of the notion of forcing and generic sets, Fund. Math. 56 (1965), 325–345.

    Google Scholar 

  • Gillman, L. and Jerison, M.: Rings of Continuous Functions, Van Nostrand, 1960.

  • Halpern, J. D.: The independence of the axiom of choice from the Boolean prime ideal theorem, Fund. Math. 55 (1964), 57–66.

    Google Scholar 

  • Halpern, J. D. and Lévy, A.: The Boolean prime ideal theorem does not imply the axiom of choice, Proc. of Symposium Pure Math. of the AMS 13 (1971), Part I, 83–134.

    Google Scholar 

  • Hartogs, F.: Über das Problem der Wohlordnung, Math. Annalen 76 (1915), 138–143.

    Google Scholar 

  • Herrlich, H.: Wann sind alle stetigen Abbildungen in Y konstant? Math. Zeitschr. 90 (1965), 152–154.

    Google Scholar 

  • Herrlich, H.: C-kompakte Räume, Math. Zeitschr. 96 (1967), 228–255.

    Google Scholar 

  • Jech, T. J.: The Axiom of Choice, North-Holland, Amsterdam, 1973.

    Google Scholar 

  • Kelley, J. L.: The Tychonoff product theorem implies the axiom of choise, Fund. Math. 37 (1950), 75–76.

    Google Scholar 

  • Kennison, J. F.: Reflective functors in general topology and elsewhere, Trans. Amer. Math. Soc. 118 (1965), 303–315.

    Google Scholar 

  • Krull, W.: Die Idealtheorie in Ringen ohne Endlichkeitsbedingungen, Mathem. Annalen 101 (1929), 729–744.

    Google Scholar 

  • Läuchli, H.: Auswahlaxiom in der Algebra, Commentarii Math. Helvetici 37 (1962–63), 1–18.

    Google Scholar 

  • Łoś, J. and Ryll-Nardzewski, C.: On the application of Tychonoff's theorem in mathematical proofs, Fund. Math. 38 (1951), 233–237.

    Google Scholar 

  • Łoś, J. and Ryll-Nardzewski, C.: Effectiveness of the representation theory for Boolean algebras, Fund. Math. 41 (1955), 49–56.

    Google Scholar 

  • Moore, G. H.: Zermelo's Axiom of Choice. Its Origins, Developments and Influence, Springer, New York, 1982.

    Google Scholar 

  • Mrówka, S.: Compactness and product spaces, Colloq. Math. 7 (1959), 19–22.

    Google Scholar 

  • Mycielski, J.: Two remarks on Tychonoff's product theorem, Bull. Acad. Polon. Sci. Sér. Sci. Math., Astr. Phys. 12 (1964), 439–441.

    Google Scholar 

  • Rubin, H. and Rubin, J. E.: Equivalents of the Axiom of Choice II, North Holland, Amsterdam, 1985.

    Google Scholar 

  • Rubin, H. and Scott, D.: Some topological theorems equivalent to the Boolean prime ideal theorem, Bull. Amer. Math. Soc. 60 (1954), 389.

    Google Scholar 

  • Salbany, S.: On compact* spaces and compactifications, Proc. Amer. Math. Soc. 45 (1974), 274–280.

    Google Scholar 

  • Sierpiński, W.: L'axiome de M. Zermelo et son rôle dans la théorie des ensembles et l'analyse, Bull. l'Acad. Sci. Cracovie Cl. Sci. Math. Sér. A (1918), 97–152.

  • Solovay, R. M.: A model of set theory in which all sets of reals are Lebesgue measurable, Ann. Math. 92 (1970), 1–56.

    Google Scholar 

  • Stone, M. H.: The theory of representations for Boolean algebras, Trans. Amer. Math. Soc. 40 (1936), 37–111.

    Google Scholar 

  • Stone, M. H.: Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), 375–481.

    Google Scholar 

  • Tarski, A.: Ein Überdeckungssatz für endliche Mengen, Fund. Math. 30 (1938), 156–163.

    Google Scholar 

  • Tychonoff, S.: Über die topologische Erweiterung von Räumen, Mathem. Annalen 105 (1930), 544–561.

    Google Scholar 

  • Tychonoff, S.: Ein Fixpunktsatz, Mathem. Annalen 111 (1935), 767–776.

    Google Scholar 

  • Ward, L. E.: A weak Tychonoff theorem and the axiom of choice, Proc. Amer. Math. Soc. 13 (1962), 757–758.

    Google Scholar 

  • Zermelo, E.: Beweis, daß jede Menge wohlgeordnet werden kann, Mathem. Annalen 59 (1904), 514–516.

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Horst Herrlich

Authors
  1. Horst Herrlich
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

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

Rights and permissions

Reprints and permissions

About this article

Cite this article

Herrlich, H. Compactness and the axiom of choice. Appl Categor Struct 4, 1–14 (1996). https://doi.org/10.1007/BF00124110

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00124110

Mathematics Subject Classifications (1991)

Key words

Search

Navigation