The topological classification of spaces of metrics with the uniform convergence topology

TL;DR

This paper classifies the topological structure of spaces of bounded pseudometrics, metrics, and admissible metrics on metrizable spaces based on properties like compactness and density, revealing their homeomorphism types.

Contribution

It provides a comprehensive topological classification of spaces of metrics and pseudometrics on metrizable spaces, extending known results to various cases based on compactness and density.

Findings

PM(X) homeomorphic to Hilbert spaces depending on X's properties

M(X) and AM(X) are homeomorphic to Hilbert spaces in specific cases

Topological types vary with compactness, separability, and density of X

Abstract

For a metrizable space $X$ of density $\kappa$, let $PM(X)$ be the space of continuous bounded pseudometrics on $X$ endowed with the uniform convergence topology. In this paper, its topology shall be classified as follows: (i) If $X$ is finite, then $PM(X)$ is homeomorphic to $\{0\}$ when $X$ is a singleton, and then $PM(X)$ is homeomorphic to $[0,1]^{\kappa(\kappa - 1)/2 - 1} \times [0,1)$ when $\kappa > 1$; (ii) If $X$ is infinite and generalized compact, then $PM(X)$ is homeomorphic to the Hilbert space $\ell_2(2^{< \kappa})$ of density $2^{< \kappa}$; (iii) If $X$ is not generalized compact, then $PM(X)$ is homeomorphic to the Hilbert space $\ell_2(2^\kappa)$ of density $2^\kappa$. Furthermore, letting $M(X)$ and $AM(X)$ be the spaces of continuous bounded metrics and bounded admissible metrics on $X$ with the subspace topology of $PM(X)$ respectively, we will recognize their topological types as follows: (iv) If $X$ is infinite and compact, then $M(X) (= AM(X))$ is homeomorphic to the separable Hilbert space $\ell_2$; (v) In the case that $X$ is not compact, $M(X)$ is homeomorphic to the Hilbert space $\ell_2(2^{\aleph_0})$ if $X$ is $\sigma$-compact, and moreover $AM(X)$ is also homeomorphic to the Hilbert space $\ell_2(2^{\aleph_0})$ if $X$ is separable locally compact.