Isometries between spaces of metrics
Katsuhisa Koshino

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 , denote by the space of continuous bounded pseudometrics on , and denote by the one of continuous bounded admissible metrics on , the both of which are equipped with the sup-norm . Let be the subspace of satisfying the following: \begin{itemize} \item for every , there exists a compact subset such that if , then . \end{itemize} Moreover, set and let be or . In this paper, we shall prove the Banach-Stone type theorem on spaces of metrics, that is, for metrizable spaces and , the following are equivalent: \begin{enumerate} \item and …
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology
