Recognizing the topologies of spaces of metrics with the topology of uniform convergence
Volume 70 / 2022
Bulletin Polish Acad. Sci. Math. 70 (2022), 165-171
MSC: Primary 54C35; Secondary 57N20, 54E35, 54E40, 54E45, 52A07.
DOI: 10.4064/ba220523-18-4
Published online: 4 May 2023
Abstract
Given a metrizable space $X$ of density $\kappa $, we study the topological structure of the space $PM(X)$ of continuous bounded pseudometrics on $X$, which is endowed with the topology of uniform convergence. We prove that $PM(X)$ is homeomorphic to $[0,1)^{\kappa (\kappa - 1)/2}$ if $X$ is finite, to $\ell _2(2^{ \lt \kappa })$ if $X$ is infinite and generalized compact, and to $\ell _2(2^\kappa )$ if $X$ is not generalized compact. We also show that for an infinite $\sigma $-compact metrizable space $X$, the space $M(X) \subset PM(X)$ of continuous bounded metrics on $X$ and the space $AM(X) \subset M(X)$ of bounded admissible metrics on $X$ are homeomorphic to $\ell _2$ if $X$ is compact, and to $\ell _\infty $ if $X$ is not compact.
