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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Weak axioms of choice for metric spaces


Authors: Kyriakos Keremedis and Eleftherios Tachtsis
Journal: Proc. Amer. Math. Soc. 133 (2005), 3691-3701
MSC (2000): Primary 03E25, 54A35, 54D65, 54D70, 54E35, 54E50, 54E99
Published electronically: June 3, 2005
MathSciNet review: 2163609
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Abstract: In the framework of ZF, i.e., Zermelo-Fraenkel set theory without the axiom of choice AC, we show that if the family of all non-empty, closed subsets of a metric space $(X,d)$ has a choice function, then so does the family of all non-empty, open subsets of $X$. In addition, we establish that the converse is not provable in ZF.

We also show that the statement ``every subspace of the real line $\mathbb{R} $ with the standard topology has a choice function for its family of all closed, non-empty subsets" is equivalent to the weak choice form ``every continuum sized family of non-empty subsets of reals has a choice function".


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Additional Information

Kyriakos Keremedis
Affiliation: Department of Mathematics, University of the Aegean, Karlovassi 83200, Samos, Greece
Email: kker@aegean.gr

Eleftherios Tachtsis
Affiliation: Department of Statistics and Actuarial Science, University of the Aegean, Karlo- vassi 83200, Samos, Greece
Email: ltah@aegean.gr

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07970-0
PII: S 0002-9939(05)07970-0
Keywords: Axiom of choice, weak axioms of choice, Loeb metric spaces, selective metric spaces, complete metric spaces, separable metric spaces, second countable metric spaces
Received by editor(s): May 29, 2004
Received by editor(s) in revised form: August 17, 2004
Published electronically: June 3, 2005
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.