2025/01/14 by Katsuhisa Koshino, Koshino, Katsuhisa
Mathematics · #54C35 #54E35 #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #Homotopy and Cohomology in Algebraic Topology #Primary 46B04 #Secondary 46E15
paper · pdf · doi:10.48550/arxiv.2501.08030
openalex publication_date 2025/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ‖⋅‖. Let \rm Pc(Z) be the subspace of \rm AM(Z) satisfying the following: \beginitemize \item for every d ∈ \rm Pc(Z), there exists a compact subset K ⊂ Z such that if d(x,y) = ‖d‖, then x, y ∈ K. \enditemize Moreover, set \rm Pp(Z) = \d ∈ \rm AM(Z) | there only exists \z,w\ ⊂ Z 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: \beginenumerate \item X and Y are homeomorphic; \item there exists a surjective isometry T : \rm PM(X) → \rm PM(Y) with T(\rm M(X)) = \rm M(Y); \item there exists a surjective isometry T : \rm AM(X) → \rm AM(Y) with T(\rm M(X)) = \rm M(Y); \item there exists a surjective isometry T : \rm M(X) → \rm M(Y). \endenumerate Then for each surjective isometry T : \rm PM(X) → \rm PM(Y) with T(\rm M(X)) = \rm M(Y), there is a homeomorphism ϕ: Y → X such that for any d ∈ \rm PM(X) and for any x, y ∈ 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.