Some Notions of Separability of Metric Spaces in $\mathbf {ZF}$ and Their Relation to Compactness

Kyriakos Keremedis¹

¹ Department of Mathematics University of the Aegean Karlovassi, Samos 83200, Greece

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 64 (2016) no. 2-3, pp. 109-136

Résumé

In the realm of metric spaces we show in $\mathbf {ZF}$ that: (1) Quasi separability (a metric space $\mathbf {X}=(X,d)$ is quasi separable iff $\mathbf {X}$ 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) $\omega $-quasi separability (a metric space $\mathbf {X}=(X,d)$ is $\omega $-quasi separable iff $\mathbf {X}$ 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 $\mathbf {CAC}$.

Détail

DOI : 10.4064/ba8087-12-2016

Keywords: realm metric spaces mathbf quasi separability metric space mathbf quasi separable mathbf has dense subset which expressible countable union finite sets weakest property under which limit point compact metric space compact omega quasi separability metric space mathbf omega quasi separable mathbf has dense subset which expressible countable union countable sets property under which countably compact metric space compact statement every totally bounded metric space separable does imply countable choice axiom mathbf cac