On HČ-completeness and Kǎtetov extensions

Abstract

We say that a Hausdorff topological space X is HČ-complete if there exists a countable family {An: n∈N} of open covers of the space X with the following property: if F is a centered family of closed subsets of X satisfying that for each n<ω there exist Fn∈F and An∈An such that Fn⊂An, then ⋂F is non-empty. We study the HČ-completeness in the Katětov extension κX of a Hausdorff space X. We determine the Katětov extension of a free topological sum and the Katětov extension of a product. Moreover we show: (1) if X is HČ-complete with a dense subset constituted by isolated points, then κX is HČ-complete, (2) for every countably compact HČ-complete space, κX is HČ-complete, and (3) if X does not have isolated points and κX is HČ-complete, then X must be feebly compact.