Weak axioms of choice for metric spaces
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- by Kyriakos Keremedis and Eleftherios Tachtsis PDF
- Proc. Amer. Math. Soc. 133 (2005), 3691-3701 Request permission
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".References
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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
- MR Author ID: 657401
- Email: ltah@aegean.gr
- 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.
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 133 (2005), 3691-3701
- MSC (2000): Primary 03E25, 54A35, 54D65, 54D70, 54E35, 54E50, 54E99
- DOI: https://doi.org/10.1090/S0002-9939-05-07970-0
- MathSciNet review: 2163609