On pseudocompactness and light compactness of metric spaces in
Volume 66 / 2018
Abstract
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.