Extending proper metrics

DOI10.1016/J.TOPOL.2022.108387zbMATH Open1508.54008arXiv2207.12905MaRDI QIDQ6406127

Yoshito Ishiki

Publication date: 26 July 2022

Abstract: We first prove a version of Tietze-Urysohn's theorem for proper functions taking values in non-negative real numbers defined on $sigma$-compact locally compact Hausdorff spaces. As its application, we prove an extension theorem of proper metrics, which states that if $X$ is a $sigma$-compact locally compact space, $A$ is a closed subset of $X$, and $d$ is a proper metric on $A$ that generates the same topology of $A$, then there exists a proper metric on $X$ such that $D$ generates the same topology of $X$ and $D{|}_{A^2}=d$. Moreover, if $A$ is a proper retraction, we can choose $D$ so that $(A,d)$ is quasi-isometric to $(X,D)$. We also show analogues of theorems explained above for ultrametric spaces.

Mathematics Subject Classification ID

Metric spaces, metrizability (54E35) Special maps on topological spaces (open, closed, perfect, etc.) (54C10) Compact (locally compact) metric spaces (54E45)

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