Extending proper metrics

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.

15 pages. This paper is published in Topology and its Applications

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