Moore-closed spaces, completeness and centered bases

Abstract

A 'space S is Moore-closed iff it is a Moore space which is a closed subspace of every Moore space including S. In this paper it is shown that there exist noncompact Moore-closed spaces and that Moore-closed spaces can be characterized as being semicomplete in a very strong sense. Related characterizations of compactness are given. Some of these results establish strong similarities and distinctions among Moore spaces, metric spaces and spaces having a base of countable order. The notions of centered base and complete centered base are introduced, and large classes of spaces are shown to have such bases.