Isometries between spaces of metrics

TL;DR

The paper extends the Banach-Stone paradigm to spaces of metrics on metrizable spaces by proving that $X$ and $Y$ are homeomorphic if and only if there is a surjective isometry between their metric spaces that preserves the distinguished class $\mathrm{M}$, with such an isometry realized by a unique (except in the $2$-point case) homeomorphism $\phi$ via $T(d)(x,y)=d(\phi(x),\phi(y))$. It introduces and analyzes the metric spaces $\text{PM}(Z)$, $\text{AM}(Z)$, $\text{Pc}(Z)$, and $\text{Pp}(Z)$, leveraging density and an approximation/extension framework to deduce structural information about isometries. A key component is the peaking-function argument, which, together with a doubleton (two-point) analysis and finite-intersection properties, yields a bijection on doubletons and ultimately a global homeomorphism implementing the isometry. The results unify compact and non-compact settings and provide a canonical representation of isometries between spaces of metrics, with implications for understanding how metric-structure spaces encode topological information.

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 $φ: Y \to X$ such that for any $d \in {\rm PM}(X)$ and for any $x, y \in Y$, $T(d)(x,y) = d(φ(x),φ(y))$. Except for the case where the cardinality of $X$ or $Y$ is equal to $2$, the homeomorphism $φ$ can be chosen uniquely.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Lemma 2.2
• Theorem 2.3
• Proposition 2.4
• proof
• Lemma 3.1
• Proposition 3.2
• proof