Extending proper metrics

TL;DR

This paper proves a version of Tietze-Urysohn's theorem for proper functions on certain topological spaces and applies it to extend proper metrics from closed subsets to entire spaces, including ultrametric cases.

Contribution

It introduces a new extension theorem for proper metrics on $\sigma$-compact locally compact spaces, generalizing previous results and including ultrametric analogues.

Findings

Extended proper metrics from closed subsets to entire spaces.

Established a proper retraction-based quasi-isometry for metrics.

Proved a Tietze-Urysohn type theorem for proper functions.

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.