A characterization of metacompactness in terms of filters (Q1407443)

A topological space \(X\) is defined to be metacompact if every covering of \(X\) by open sets has a point finite open refinement. The authors call a collection of subsets \(\Omega\) of a set \(X\) point dominating if each \(x \in X\) is a member of all but finitely many elements of \(\Omega\). Also a filter on a space is said to be of type \(\mathcal M\) if every point dominating subcollection of the filter has nonempty adherence. It is proved that a topological space is metacompact if and only if every filter of type \(\mathcal M\) on the space has nonempty adherence. This characterization is used to give new proofs for results like: `a countably compact metacompact space is compact', `a closed subspace of a metacompact space is metacompact' and `the product of a compact space and a metacompact space is metacompact'.