Weak axioms of choice for metric spaces

Keremedis, Kyriakos; Tachtsis, Eleftherios

doi:10.1090/S0002-9939-05-07970-0

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".

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