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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.

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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