Concerning ultraparacompact spaces.

Abstract

In this interesting paper a new class of ultraparacompact spaces, the class of so-called (P*) spaces, is introduced and investigated. A zero-dimensional Hausdorff space is called a (P*) space if for every open cover and for every clopen base such that every finite union of elements of the base can be partitioned into elements of the base, there exists a discrete refinement consisting of members of the base. For example, a zero-dimensional dense subspace of Rn is a (P*) space. The authors provide a characterization of the (P*) property and show that every ultraparacompact C-scattered space satisfies property (P*). At the end of the paper some open questions are raised.