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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.

On Sequential Compactness and Related Notions of Compactness of Metric Spaces in ZF

Keremedis K.

Bulletin of the Polish Academy of Sciences. Mathematics

2016

Vol. 64, no. 1

29--46

We show that: (i) If every sequentially compact metric space is countably compact then for every infinite set X, [X]<ω is Dedekind-infinite. In particular, every infinite subset of R is Dedekind-infinite. (ii) Every sequentially compact metric space is compact iff every sequentially compact metric space is separable. In addition, if every sequentially compact metric space is compact then: every infinite set is Dedekind-infinite, the product of a countable family of compact metric spaces is compact, and every compact metric space is separable. (iii) The axiom of countable choice implies that every sequentially bounded metric space is totally bounded and separable, every sequentially compact metric space is compact, and every uncountable sequentially compact, metric space has size |R|. (iv) If every sequentially bounded metric space is totally bounded then every infinite set is Dedekind-infinite. (v) The statement: “Every sequentially bounded metric space is bounded” implies the axiom of countable choice restricted to the real line. (vi) The statement: “For every compact metric space X either |X| ≤ |R|, or |R| ≤ |X|” implies the axiom of countable choice restricted to families of finite sets. (vii) It is consistent with ZF that there exists a sequentially bounded metric space whose completion is not sequentially bounded. (viii) The notion of sequential boundedness of metric spaces is countably productive.