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Extending proper metrics

Yoshito Ishiki
Jul 2022
摘要
We first prove Tietze-Urysohn's theorem for proper functions taking values innon-negative real numbers defined on $\sigma$-compact locally compact Hausdorffspaces. As its application, we prove an extension theorem of proper maps. Let$X$ be a $\sigma$-compact locally compact space. Let $A$ be a closed subset$A$. Let $d$ be 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 sametopology of $X$ and $D|_{A^{2}}=d$. If $A$ is a proper retraction, We canchoose $D$ so that $(A, d)$ is quasi-isometric to $(X, D)$. We also showanalogues of theorems explained above for ultrametric spaces.
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