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axiom of choice; compact space; countably compact space; totally bounded space; Lindelöf space; separable space; second countable metric space
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.
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[10] Keremedis K.: On metric spaces where continuous real valued functions are uniformly continuous in $\mathbf {ZF}$. Topology Appl. 210 (2016), 366–375. DOI 10.1016/j.topol.2016.07.021
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