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.
Categories: math.FA math.GN
