Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-19T20:58:51.962Z Has data issue: false hasContentIssue false
  • English
  • Français

Products of some special compact spaces and restricted forms of AC

Published online by Cambridge University Press:  12 March 2014

Kyriakos Keremedis  and
Eleftherios Tachtsis
Kyriakos Keremedis
Affiliation:
University of the Aegean, Department of Mathematics, Karlovassi, Samos 83200, Greece. E-mail: kker@aegean.gr
Eleftherios Tachtsis
Affiliation:
University of the Aegean, Department of Statistics and Actuarial-Financial Mathematics, Karlovassi, Samos 83200, Greece. E-mail: ltah@aegean.gr
Get access Rights & Permissions [Opens in a new window]

Abstract

We establish the following results:

1. In ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC), for every set I and for every ordinal number α ≥ ω, the following statements are equivalent:

(a) The Tychonoff product of ∣α∣ many non-empty finite discrete subsets of I is compact.

(b) The union of ∣α∣ many non-empty finite subsets of I is well orderable.

2. The statement: For every infinite set I, every closed subset of the Tychonoff product [0, 1]Iwhich consists offunctions with finite support is compact, is not provable in ZF set theory.

3. The statement: For every set I, the principle of dependent choices relativised to I implies the Tychonoff product of countably many non-empty finite discrete subsets of I is compact, is not provable in ZF0 (i.e., ZF minus the Axiom of Regularity).

4. The statement: For every set I, every ℵ0-sized family of non-empty finite subsets of I has a choice function implies the Tychonoff product of ℵ0many non-empty finite discrete subsets of I is compact, is not provable in ZF0.

Keywords

Axiom of Choiceweak Axioms of ChoiceTychonoff productscompactnesspermutation model
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 3 , September 2010 , pp. 996 - 1006
Copyright
Copyright © Association for Symbolic Logic 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Brunner, N., Amorphe Potenzen kompakter Räume, Archiv für Mathematische Logik und Grundlagenforschung, vol. 24 (1984), pp. 119–135.CrossRefGoogle Scholar
[2]De la Cruz, O., Hall, E., Howard, P., Keremedis, K., and Rubin, J. E., Products of compact spaces and the Axiom of Choice, Mathematical Logic Quarterly, vol. 48 (2002), pp. 508–516.3.0.CO;2-#>CrossRefGoogle Scholar
[3]Cruz, O. Dela, Hall, E., Howard, P., Keremedis, K., and Rubin, J. E., Products of compact spaces and the Axiom of Choice II, Mathematical Logic Quarterly, vol. 49 (2003), pp. 57–71.CrossRefGoogle Scholar
[4]De la Cruz, O., Hall, E., Howard, P., Rubin, J. E., and Stanley, A., Definitions of compactness and the axiom of choice, this Journal, vol. 67 (2002), pp. 143–161.Google Scholar
[5]Good, C. and Tree, I., Continuing horrors of topology without choice, Topology and its Applications, vol. 63 (1995), pp. 79–90.CrossRefGoogle Scholar
[6]Howard, P., Definitions of compact, this Journal, vol. 55 (1990), pp. 645–655.Google Scholar
[7]Howard, P., Keremedis, K., Rubin, J. E., and Stanley, A., Compactness in countable Tychonoff products and choice, Mathematical Logic Quarterly, vol. 46 (2000), pp. 3–16.3.0.CO;2-E>CrossRefGoogle Scholar
[8]Howard, P. and Rubin, J. E., Consequences of the Axiom of Choice, Mathematical Surveys and Monographs, vol. 59, American Mathematical Society, Providence RI, 1998.CrossRefGoogle Scholar
[9]Jech, T. J., The Axiom of Choice, North-Holland, Amsterdam, 1973, reprint: Dover Publications, Inc., Mineola, New York, 2008.Google Scholar
[10]Kelley, J. L., The Tychonoff product theorem implies the Axiom of Choice, Fundamenta Mathematicae, vol. 37 (1950), pp. 75–76.CrossRefGoogle Scholar
[11]Keremedis, K., Tychonoff products of two-element sets and some weakenings of the Boolean prime ideal theorem, Bulletin of the Polish Academy of Sciences. Mathematics, vol. 53 (2005), no. 4, pp. 349–359.CrossRefGoogle Scholar
[12]Loeb, P. A., A new proof of the Tychonoff theorem, American Mathematics Monthly, vol. 72 (1965), pp. 711–717.CrossRefGoogle Scholar
[13]Morillon, M., Countable choice and compactness, Topology and its Applications, vol. 155 (2008), no. 10, pp. 1077–1088.CrossRefGoogle Scholar
[14]Morillon, M., Notions of compactness for special subsets of ℝI and some weak forms of the Axiom of Choice, communicated manuscript.Google Scholar
[15]Morillon, M., Synthèse, preprint, (see http://personnel.univ-reunion.fr/mar).Google Scholar
[16]Mycielski, J., Two remarks on Tychonoff's product theorem, Bulletin de l'Academie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 12 (1964), pp. 439–441.Google Scholar
[17]Rubin, H. and Scott, D., Some topological theorems equivalent to the Boolean prime ideal theorem, Bulletin of the American Mathematical Society, vol. 60 (1954), p. 389.Google Scholar