  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 $X$.
