On Weierstrass compact pseudometric spaces and a weak form of the axiom of choice
\textit{H. L. Bentley} and \textit{H. Herrlich} [Topology Appl. 85, No. 1-3, 153-164 (1998; Zbl 0922.03068)] investigated in ZF (= Zermelo-Fraenkel set theory without the Axiom of Choice) the relations between various compactness notions for pseudometric spaces. The question, under which set-theoretical condition every Weierstrass-compact (i.e., every infinite subset has an accumulation point) pseudometric space is compact, remained open. In the paper under review the author provides an elegant solution for this problem by showing that in ZF the following conditions are equivalent: Every Weierstrass-compact pseudometric space is compact. The Countable Axiom of Multiple Choice holds. Every Weierstrass-compact pseudometric space is countably compact. Every compact pseudometric space contains a dense subset which is a countable union of finite sets. (In the above mentioned paper by Bentley and Herrlich it has been shown that the Axiom of Countable Choice holds iff every compact pseudometric space is separable).
scientific article; zbMATH DE number 1538677