On isometric universality of spaces of metrics

TL;DR

This paper investigates the conditions under which spaces of metrics on various types of spaces are universal for classes of metric spaces, revealing both universal and non-universal cases based on properties of the underlying space.

Contribution

It establishes new results on the isometric universality of metric spaces of metrics for different classes of base spaces, including bounded, separable, compact, and discrete spaces.

Findings

Space of bounded metrics on Z is universal for bounded metric spaces.

Space of metrics on an infinite discrete space is universal for all separable metric spaces.

Non-compact or uncountable compact Z yields universal spaces for all compact metric spaces.

Abstract

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 $Z$.