Isometries between spaces of metrics

TL;DR

This paper establishes a Banach-Stone type theorem characterizing homeomorphic spaces via isometries between spaces of bounded continuous metrics and pseudometrics, revealing a deep connection between metric space structure and isometric transformations.

Contribution

It proves that homeomorphism between metrizable spaces is equivalent to the existence of certain surjective isometries between their metric spaces, extending classical Banach-Stone results to spaces of metrics.

Findings

Homeomorphic spaces correspond to isometric metric spaces.

Isometries induce homeomorphisms between underlying spaces.

Uniqueness of the homeomorphism except for 2-point spaces.

Abstract

Given a metrizable space $Z$, denote by ${\rm PM}(Z)$ the space of continuous bounded pseudometrics on $Z$, and denote by ${\rm AM}(Z)$ the one of continuous bounded admissible metrics on $Z$, the both of which are equipped with the sup-norm $\|\cdot\|$. Let ${\rm Pc}(Z)$ be the subspace of ${\rm AM}(Z)$ satisfying the following: \begin{itemize} \item for every $d \in {\rm Pc}(Z)$, there exists a compact subset $K \subset Z$ such that if $d(x,y) = \|d\|$, then $x, y \in K$. \end{itemize} Moreover, set $${\rm Pp}(Z) = \{d \in {\rm AM}(Z) \mid \text{ there only exists } \{z,w\} \subset Z \text{ such that } d(z,w) = \|d\|\},$$ and let ${\rm M}(Z)$ be ${\rm Pc}(Z)$ or ${\rm Pp}(Z)$. In this paper, we shall prove the Banach-Stone type theorem on spaces of metrics, that is, for metrizable spaces $X$ and $Y$, the following are equivalent: \begin{enumerate} \item $X$ and $Y$ are homeomorphic; \item there exists a surjective isometry $T : {\rm PM}(X) \to {\rm PM}(Y)$ with $T({\rm M}(X)) = {\rm M}(Y)$; \item there exists a surjective isometry $T : {\rm AM}(X) \to {\rm AM}(Y)$ with $T({\rm M}(X)) = {\rm M}(Y)$; \item there exists a surjective isometry $T : {\rm M}(X) \to {\rm M}(Y)$. \end{enumerate} Then for each surjective isometry $T : {\rm PM}(X) \to {\rm PM}(Y)$ with $T({\rm M}(X)) = {\rm M}(Y)$, there is a homeomorphism $\phi : Y \to X$ such that for any $d \in {\rm PM}(X)$ and for any $x, y \in Y$, $T(d)(x,y) = d(\phi(x),\phi(y))$. Except for the case where the cardinality of $X$ or $Y$ is equal to $2$, the homeomorphism $\phi$ can be chosen uniquely.