Topological characterizations of ω μ-metrizable spaces

Abstract

This paper is a detailed elaboration of a talk given by the second author at the Oxford conference in June 1989. It presents necessary and sufficient conditions for a topological space to be ω μ-metrizable (μ> 0), i.e., linearly uniformizable with uncountable uniform weight. In other words, such spaces are exactly those which can be metrized by a distance function taking its values in a totally ordered Abelian group with cofinality ω μ. (For ω μ = ω 0, we obtain characterizations of strongly zero-dimensional metric spaces, i.e., nonarchimedeanly metrizable spaces.) It turns out that (strong) suorderability and the existence of a σ-discrete (respectively ω μ- discrete) dense subspace are the most interesting properties in this respect, whenever ω μ > ω 0, or ω μ = ω 0 and dim X = 0. Therefore, a main part of the paper is devoted to the study of GO- spaces having a σ-discrete (ω μ) dense subspace (Section 3). The last section (Section 4) is concerned with the characterization of ω μ-metrizability in the realm of generalized metric spaces, in particular, by using g-functions. Since all our spaces are zero-dimensional, the paper also contributes results to this important class of spaces, in particular, to the class of nonarchimedean topological spaces.