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Notions of compactness for special subsets of ℝI and some weak forms of the axiom of choice

Published online by Cambridge University Press:  12 March 2014

Marianne Morillon
Marianne Morillon*
Affiliation:
Université de la Réunion, Parc Technologique Universitaire, Ermit, Département de Mathématiques et Informatique, Bâtiment 2, 2 Rue Joseph Wetzell, 97490 Sainte-Clotilde, France, E-mail: [email protected], URL: http://personnel.univ-reunion.fr/mar
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Abstract

We work in set-theory without choice ZF . A set is countable if it is finite or equipotent with ℕ. Given a closed subset F of [0, 1]I which is a bounded subset of ℓ 1(I) (resp. such that F ⊆ c 0(I)), we show that the countable axiom of choice for finite sets, (resp. the countable axiom of choice AC ℕ) implies that F is compact. This enhances previous results where AC ℕ (resp. the axiom of Dependent Choices) was required. If I is linearly orderable (for example I = ℝ), then, in ZF , the closed unit ball of the Hilbert space ℓ 2 (I) is (Loeb-)compact in the weak topology. However, the weak compactness of the closed unit ball of is not provable in ZF .

Keywords

Axiom of Choiceproduct topologycompactnesssequential compactness
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 1 , March 2010 , pp. 255 - 268
Copyright
Copyright © Association for Symbolic Logic 2010

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References

REFERENCES

[1] Avilés, A., The unit ball of the Hilbert space in its weak topology, Proceedings of the American Mathematical Society, vol. 135 (2007), no. 3, pp. 833–836.Google Scholar
[2] Benyamini, Y., Rudin, M. E., and Wage, M., Continuous images of weakly compact subsets of Banach spaces, Pacific Journal of Mathematics, vol. 70 (1977), no. 2, pp. 309–324.Google Scholar
[3] Brunner, N., Products of compact spaces in the least permutation model, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 31 (1985), no. 5, pp. 441–448.Google Scholar
[4] Fossy, J. and Morillon, M., The Baire category property and some notions of compactness, Journal of the London Mathematical Society, II, vol. 57 (1998), no. 1, pp. 1–19.Google Scholar
[5] Howard, P., Keremedis, K., Rubin, J. E., and Stanley, A., Compactness in countable Tychonoff products and choice, Mathematical Logic Quarterly, vol. 46 (2000), no. 1, pp. 3–16.Google Scholar
[6] Howard, P. and Rubin, J. E., Consequences of the Axiom of Choice, vol. 59, American Mathematical Society, Providence, RI, 1998.Google Scholar
[7] Jech, T. J., The Axiom of Choice, North-Holland Publishing, Amsterdam, 1973.Google Scholar
[8] Kelley, J. L., The Tychonoff product theorem implies the Axiom of Choice, Fundamenta Mathematicae, vol. 37 (1950), pp. 75–76.Google Scholar
[9] Keremedis, K., The compactness of 2R and the Axiom of Choice, Mathematical Logic Quarterly, vol. 46 (2000), no. 4, pp. 569–571.Google Scholar
[10] 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.Google Scholar
[11] Keremedis, K. and Tachtsis, E., On Loeb and weakly Loeb Hausdorff spaces, Scientiae Mathematicae Japonicae, vol. 53 (2001), no. 2, pp. 247–251.Google Scholar
[12] Loeb, P. A., A new proof of the Tychonoff theorem, American Mathematical Monthly, vol. 72 (1965), pp. 711–717.Google Scholar
[13] Morillon, M., Countable choice and compactness, Topology andits Applications, vol. 155 (2008), no. 10, pp. 1077–1088.Google Scholar
[14] Simon, P., On continuous images of Eberlein compacts, Commentationes Mathematicae Universitatis Carolinae, vol. 17 (1976), no. 1, pp. 179–194.Google Scholar