Abstract:In the realm of metric spaces we show in {\rm ZF} that: (i) A metric space is compact if and only if it is countably compact and for every $\varepsilon > 0$, every cover by open balls of radius $\varepsilon $ has a countable subcover. (ii) Every second countable metric space has a countable base consisting of open balls if and only if the axiom of countable choice restricted to subsets of $\mathbb{R}$ holds true. (iii) A countably compact metric space is separable if and only if it is second countable.
Keywords: axiom of choice; compact space; countably compact space; totally bounded space; Lindel\"of space; separable space, second countable metric space
DOI: DOI 10.14712/1213-7243.2015.229
AMS Subject Classification: 54E35 54E45