Geodesic
Some versions of second countability of metric spaces in ZF and their role to compactness
Commentationes Mathematicae Universitatis Carolinae, Tome 59 (2018) no. 1, pp. 119-134.

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In the realm of metric spaces we show in 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.
DOI : 10.14712/1213-7243.2015.229
Classification : 54E35, 54E45
Keywords: axiom of choice; compact space; countably compact space; totally bounded space; Lindelöf space; separable space; second countable metric space 
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Keremedis, Kyriakos. Some versions of second countability of metric spaces in ZF and their role to compactness. Commentationes Mathematicae Universitatis Carolinae, Tome 59 (2018) no. 1, pp. 119-134. doi : 10.14712/1213-7243.2015.229. http://geodesic.mathdoc.fr/articles/10.14712/1213-7243.2015.229/

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