## Kyriakos Keremedis

*Some versions of second countability of metric spaces in ZF and their role to compactness*

Comment.Math.Univ.Carolin. 59,1 (2018) 119-134.**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

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