Simultaneous generalizations of paracompactness and semi-stratifiability

Abstract

In 1951, Bing introduced the concept of a topological space in which given an open cover U there is associated a sequence a_1, a_2, . . . of discrete collections of closed sets such that Union from n=1 to infinity of a_n refines U. Having established that developable spaces have this property, he applied this result improving that a collectionwise normal Moore space is metrizable. Bing further established that a collectionwise normal space satisfying the above "discrete closed refinement" condition is paracompact, thereby providing an alternate verification that metric spaces are paracompact. In 1958, McAuley introduced the terms F_sigma-screenable for a space satisfying Bing's conditions, and in addition, proved that semi-metric spaces are F_sigma-screenable. It is the purpose of this paper to investigate F_sigma-screenable spaces further, to formulate several new generalizations of such spaces, to determine properties and relationships among these spaces, and to relate them to other, more well-known spaces. In particular, it is shown that paracompact spaces are F_sigma-screenable. This fact, together with a recent result due to Creede, establishes F_sigma-screenable spaces as a simultaneous generalization of paracompact and of semi-stratifiable spaces. It is shown that F_sigma-screenable spaces are characterized by the property that every open cover has a sigma-locally finite closed refinement, a seemingly much weaker condition. In addition to its intrinsic interest, this characterization is used to establish several results concerning F_sigma-screenable spaces. Spaces in which open covers have either sigma-closure-preserving closed refinements (a sigma-CP space) or sigma-compact-finite closed refinements (a sigma-CF space) are shown to be generalizations of F_sigma-screenability, and to imply F_sigma-screenability in metacompact and k-spaces, respectively. Finally, several generalizations of locally finite collections are considered.