A relation between perfect separability, completeness, and normality in semi-metric spaces

Abstract

l Introduction* This paper proves that a regular semi-metric 1 topological space S may have such properties as hereditary separability, collectionwise normality [1], paracompactness However, if S is strongly complete, then hereditary separability implies perfect separability It has been proved [1; 12] that a regular developable topological space (Moore space) is metrizable provided that it is perfectly separable. Thus, a regular semi-metric topological space may be far removed from a Moore space contrary to a result announced by C. W. Vickery The notion of p-separability due to Frechet is generalized and a question raised by W. A. Wilson [14, p. 336] is answered in the affirmative. Throughout this paper, denotes a regular semi-metric topological space.