On pseudocompactness and light compactness of metric spaces in $\mathbf {ZF}$

In the realm of metric spaces we show, in the Zermelo–Fraenkel set theory $\mathbf{ZF}$, that:

(a) A metric space $\mathbf{X}=(X,d)$ is countably compact iff it is pseudocompact.

(b) Given a metric space $\mathbf{X}=(X,d),$ the following statements are equivalent:

$\hskip2em$(i) $\mathbf{X}$ is lightly compact (every locally finite family of open sets is finite).

$\hskip2em$(ii) Every locally finite family of subsets of $\mathbf{X}$ is finite.

$\hskip2em$(iii) Every locally finite family of closed subsets of $\mathbf{X}$ is finite.

$\hskip2em$(iv) Every pairwise disjoint, locally finite family of subsets of $\mathbf{X} $ is finite.

$\hskip2em$(v) Every pairwise disjoint, locally finite family of closed subsets of $% \mathbf{X}$ is finite.

$\hskip2em$(vi) Every locally finite, pairwise disjoint family of open subsets of $% \mathbf{X}$ is finite.

$\hskip2em$(vii) Every locally finite open cover of $\mathbf{X}$ has a finite subcover.

(c) For every infinite set $X$, the powerset $\mathcal{P}(X)$ of $X$ has a countably infinite subset iff every countably compact metric space is lightly compact.