K*-Metrizable spaces and their applications

In this paper, we introduce and study a new class of generalized metric spaces, which we call k*-metrizable spaces, and suggest various applications of such spaces in topological algebra, functional analysis, and measure theory. By definition, a Hausdorff topological space X is k*-metrizable if X is the image of a metrizable space M under a continuous map f: M → X which has a section s: X → M preserving precompact sets in the sense that the image s(K) of any compact set K ⊂X has compact closure in X. © 2008 Springer Science+Business Media, Inc.