On isometric universality of spaces of metrics

A metric space $(M, d)$ is said to be universal for a class of metric spaces if all metric spaces in the class can be isometrically embedded into $(M, d)$. In this paper, for a metrizable space $Z$ possessing abundant subspaces, we first prove that the space of bounded metrics on $Z$ is universal for all bounded metric spaces (with restricted cardinality). Next, in contrast, we show that if $Z$ is an infinite discrete space, then the space of metrics on $Z$ is universal for all separable metric spaces. As a corollary of our results, if $Z$ is non-compact, or uncountable and compact, then the space of metrics on $Z$ is universal for all compact metric spaces. In addition, if $Z$ is compact and countable, then there exists a compact metric space that can not be isometrically embedded into the space of metrics on $Z$.