$k^*$-Metrizable Spaces and their Applications
  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 $X$. 
T. O. Banakh, V. I. Bogachev, A. V. Kolesnikov