Pointed Gromov-Hausdorff Topological Stability for Non-compact Metric Spaces

We combine the pointed Gromov-Hausdorff metric with the locally C0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^0$$\end{document}-distance to obtain the pointed C0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^0$$\end{document}-Gromov-Hausdorff distance between maps of possibly different non-compact pointed metric spaces. The latter is combined with Walters’s locally topological stability proposed by Lee–Nguyen–Yang, and GH-stability from Arbieto-Morales to obtain the notion of topologically GH-stable pointed homeomorphism. We give one example to show the difference between the distance when taking different base points in a pointed metric space.