Year: 2026
Type: preprint
Abstract: Let $X$ be a metrizable space, and let $\mathrm{Met}(X)$ denote the space of metrics compatible with the topology of $X$, regarded as a subspace of the space of continuous pseudometrics with the topology induced by the supremum metric. We first prove that if $Z$ is a discrete space whose cardinality is the first uncountable cardinal, then $\mathrm{Met}(Z)$ is not Borel. As a consequence, if $\mathrm{Met}(X)$ is Borel, then $X$ is separable. Combined with a theorem of Koshino, our result yields that the space of bounded compatible metrics on a metrizable space $X$ is completely metrizable if and only if $X$ is $σ$-compact. We also establish non-Archimedean analogues for spaces of ultrametrics. We further prove componentwise versions of these results, where two metrics belong to the same component if their distance is finite.
Source: arXiv (Cornell University)
Authors: Ishiki, Yoshito, Uda, Tomoki
Institutions: Tokyo Metropolitan University, Nanzan University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203862218