On lightly and countably compact spaces in ZF

Abstract

Given a topological space X = (X, T), we show in the Zermelo-Fraenkel set theory ZF that:

(i) Every locally finite family of open sets of X is finite iff every pairwise disjoint, locally finite family of open sets is finite.
(ii) Every locally finite family of subsets of X is finite iff every pairwise disjoint,
locally finite family of subsets of X is finite iff every locally finite family of closed subsets of X is finite.
(iii) The statement \every locally finite family of closed sets of X is finite" implies the proposition \every locally finite family of open sets of X is finite". The converse holds true in case X is T₄ and the countable axiom of choice holds true.

We also show:
(iv) It is relatively consistent with ZF the existence of a non countably compact T₁ space such that every pairwise disjoint locally finite family of closed subsets is finite but some locally finite family of subsets is infinite.
(v) It is relatively consistent with ZF the existence of a countably compact T₄ space including an infinite pairwise disjoint locally finite family of open (resp. closed) sets.

Mathematics Subject Classification (2010): E325, 54D30.

Keywords: Axiom of choice, countably compact, pseudocompact and lightly  compact topological spaces