Unity of compactness

Abstract

Let X be paracompact, that is, every open cover P of X admits a locally finite open refinement R covering X. Does there exist a class D of covers of X such that the paracompactness of X is equivalent to its D-compactness (each D ∈ D admits a finite subcover of X)? We address the question in a more general framework of P/R-compact versus D-compact families of subsets in a pretopological space and obtain an affirmative answer as a corollary.