On separable z-filters

Abstract

This paper investigates a topological property, intermediate between paracompactness and topological completeness, which arises from considerations in topological measure theory. A z-filter on a completely regular Hausdorff space X is said to be separable if, for each bounded continuous pseudometric d on X, there is a member of the filter which is separable in the pseudometric space determined by d. Then X is said to be separably paracompact if every separable z-filter with the countable intersection property is fixed. The important measure-theoretic consequence of separable paracompactness is indicated, and connections are drawn between this concept and known characterizations of topological completeness, realcompactness, the Lindelöf property, and paracompactness. The paper concludes with an examination of P-spaces; the question of whether a topologically complete P-space must be separably paracompact is not resolved, but certain simplications are obtained.