Extending proper metrics

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 an application, we prove an extension theorem of proper metrics, which states that if X is a Sigma-compact locally compact Hausdorff 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 D on X such that D generates the same topology of X and D|A2 = d. Moreover, if A is a proper retract, we can choose D so that (A, d) is quasi-isometric to (X, D). We also show analogues of the theorems explained above for ultrametrizable spaces.(c) 2022 Elsevier B.V. All rights reserved.