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

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 $\mathbf {CAC}$.