Mathematics > Metric Geometry
[Submitted on 26 Jul 2022 (v1), last revised 24 Dec 2022 (this version, v4)]
Title:Extending proper metrics
View PDFAbstract: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.
Submission history
From: Yoshito Ishiki [view email][v1] Tue, 26 Jul 2022 13:54:38 UTC (11 KB)
[v2] Thu, 28 Jul 2022 04:23:02 UTC (11 KB)
[v3] Wed, 10 Aug 2022 03:19:28 UTC (11 KB)
[v4] Sat, 24 Dec 2022 02:45:59 UTC (13 KB)
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