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Compact metric spaces and weak forms of the axiom of choice. (English) Zbl 0968.03057
A topological space is Loeb (weakly Loeb) if the family of the closed nonempty subsets admits a choice function (a multiple choice function which chooses nonempty finite subsets). Since these selections need not be continuous, their theory is a topic of set theory without the axiom of choice. The present paper investigates the Loeb property of compact metric spaces. By means of a permutation model the authors show that compact metric spaces need not be weakly Loeb or separable, whence their following result is nontrivial: For compact metric spaces, Loeb is equivalent to separable.

MSC:
03E25 Axiom of choice and related propositions
54A35 Consistency and independence results in general topology
54D65 Separability of topological spaces
54E35 Metric spaces, metrizability
54E45 Compact (locally compact) metric spaces
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