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

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.