Some Notions of Separability of Metric Spaces in and Their Relation to Compactness
Tom 64 / 2016
Streszczenie
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}.