Given a metrizable space Z, denote by \(\operatorname {PM}(Z)\) the space of continuous bounded pseudometrics on Z, and denote by \(\operatorname {AM}(Z)\) the one of continuous bounded admissible metrics on Z, both of which are equipped with the sup-norm \(\Vert \cdot \Vert .\) In this paper, we shall prove Banach–Stone type theorems on spaces of metrics, that is, for metrizable spaces X and Y, X and Y are homeomorphic if and only if there exists a surjective isometry \(T: \operatorname {PM}(X) \rightarrow \operatorname {PM}(Y)\) \((T: \operatorname {AM}(X) \rightarrow \operatorname {AM}(Y))\) satisfying some conditions. Then for each surjective isometry T, there is a homeomorphism \(\phi : Y \rightarrow X\) such that for any \(d \in \operatorname {PM}(X)\) 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.
