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 investigate the universality of the space of metrics on Z . Next, in contrast, we show that if Z is an infinite discrete space, then the space of metrics on Z is universal for all metric spaces having the same weight of Z . 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 .