Some generalizations of paracompactness

Abstract

An indexed point set {p_alpha} alpha in A in a topological space is said to have an accumulation point p in X if, for every neighborhood U of p there exist infinitely many alpha in A such that p_alpha in U. The purpose of this dissertation is to investigate properties of coverings of topological spaces in terms of conditions on indexed point sets chosen in a particular way from members of the covering. Two spaces are defined in terms of refinements of open coverings and the interaction between these notions and the concepts of metric space, paracompact space, and other classes of spaces including the family of paracompact spaces is investigated. Let U = {U_alpha | alpha in A} be a family of subsets of a Hausdorff space. A family U is said to be freely homogeneous if no indexed point set {p_alpha} alpha in A, where p_alpha in U_alpha for each alpha in A, has an accumulation point. A family U is said to be weakly homogeneous if, for every subcollection {U_alpha |alpha in A'} of distinct members of U, where A' subset of A, such that, if p_alpha and q_alpha in U_alpha for each alpha in A', and if {p_alpha} alpha in A' has an accumulation point, then {q_alpha} alpha in A' has an accumulation point. A space X is said to be freely homocompact (weakly-homocompact) if every open covering of X has a freely homogeneous (weakly homogeneous) open refinement. Every paracompact space is freely homocompact; every freely homocompact space is weakly homocompact, as is every countably compact space. In addition, every freely homocompact space is metacompact. A metacompact, weakly homocompact space is freely homocompact. Free homocompactness characterizes paracompactness in q-spaces. Some characterizations of free homocompactness are given in terms of apparently weaker conditions on refinements of open covers. The property of being freely homocompact is preserved under perfect mappings and their inverses. The product of a freely homocompact space and a compact space is freely homocompact. Normal, freely homocompact spaces are countably paracompact while normal, weakly homocompact, q-spaces are collectionwise normal. A family of subsets U in a space X is said to be locally free homogeneous if, for every point p in X and every neighborhood V of p, no indexed point set of the form {p_u | p_u in U in U and U intersection [X\V] not equal to null set} has an accumulation point. A space X is freely homogeneous if and only if every open covering has a locally free homogeneous open refinement. As applications of these notions the following metrization theorems for regular spaces are presented. A space is metrizable if and only if it has a sigma-freely homogeneous base. A freely homogeneous Moore space is metrizable. A weakly homogeneous space with a uniform base is metrizable. A space is metrizable if and only if it has a locally free homogeneous base.