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Some Notions of Separability of Metric Spaces in ZF and Their Relation to Compactness

Keremedis K.

Bulletin of the Polish Academy of Sciences. Mathematics

2016

Vol. 64, no. 2/3

119--136

In the realm of metric spaces we show in ZF that: (1) Quasi separability (a metric space X = (X, d) is quasi separable iff X has a dense subset which is expressible as a countable union of finite sets) is the weakest property under which a limit point compact metric space is compact. (2) ω-quasi separability (a metric space X = (X, d) is ω-quasi separable iff X has a dense subset which is expressible as a countable union of countable sets) is a property under which a countably compact metric space is compact. (3) The statement “Every totally bounded metric space is separable” does not imply the countable choice axiom CAC.