$k^*$-Metrizable Spaces and their Applications

TL;DR

This paper introduces $k^*$-metrizable spaces, a new class of generalized metric spaces, and explores their applications in topological algebra, functional analysis, and measure theory.

Contribution

It defines $k^*$-metrizable spaces and investigates their properties and applications across various areas of mathematics.

Findings

$k^*$-metrizable spaces generalize metric spaces.

Applications in topological algebra, functional analysis, measure theory.

Characterization of $k^*$-metrizable spaces via continuous maps with sections.

Abstract

In this paper we introduce and study so-called $k^*$-metrizable spaces forming a new class of generalized metric spaces, and display 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 $f:M\to X$ having a section $s:X\to M$ that preserves precompact sets in the sense that the image $s(K)$ of any compact set $K\subset X$ has compact closure in $X$.