Mesocompact and sequentially mesocompact spaces

Abstract

A topological space is said to have the K-condition if it is a k-space. For instance, the locally compact spaces and the first countable spaces have the K-condition. The notions of paracompact space and k-space both have the distinction of being simultaneous generalizations of the concepts of compact space and metric space. The purpose of this dissertation is to present some of the interactions between these notions. Specifically, the paracompact spaces with the K-condition are characterized. To this end, four generalizations of the notion of locally finite family are introduced. Definition: A family (F_alpha : alpha in A) of subsets of a Hausdorff space X is said to be compact-finite (cs-finite) if for each compact set K in X (for each convergent sequence {p_i} in X) K intersection F_alpha ? ¿ (F_alpha intersection {p_i} ? ¿) for at most finitely many alpha in A. Definition: A family (F_alpha: alpha in A} of subsets of a Hausdorff space X is said to be strongly compact-finite (strongly cs-finite) if {cl(F_alpha): alpha in A} is compact-finite (cs-finite). Definition: A space X is said to be strongly mesocompact (mesocompact, strongly sequentially mesocompact, sequentially mesocompact) if every open covering of X has a strongly compact-finite (compact-finite, strongly cs-finite, cs-finite) open refinement. The main results in the characterization of the para¿compact spaces with the K-condition are indicated in the following diagram. [Diagram] As applications of these notions the following metrization theorems are presented. Theorem: A space X is metrizable if and only if X is regular and has a sigma-cs-finite base. Theorem: A sequentially mesocompact Moore space is metrizable. A relationship between the weak topology of Whitehead and the weak topology of Morita is established in the general setting of k-collections.