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On pseudocompactness and light compactness of metric spaces in ZF

Keremedis K.

Bulletin of the Polish Academy of Sciences. Mathematics

2018

Vol. 66, no. 2

99--113

In the realm of metric spaces we show, in the Zermelo-Fraenkel set theory ZF, that: (a) A metric space X = (X, d) is countably compact iff it is pseudocompact. (b) Given a metric space X = (X, d); the following statements are equivalent: (i) X is lightly compact (every locally finite family of open sets is finite). (ii) Every locally finite family of subsets of X is finite. (iii) Every locally finite family of closed subsets of X is finite. (iv) Every pairwise disjoint, locally finite family of subsets of X is finite. (v) Every pairwise disjoint, locally finite family of closed subsets of X is finite. (vi) Every locally finite, pairwise disjoint family of open subsets of X is finite. (vii) Every locally finite open cover of X has a finite subcover. (c) For every infinite set X, the powerset P(X) of X has a countably infinite subset iff every countably compact metric space is lightly compact.