Pointed Gromov-Hausdorff Topological Stability for non-compact metric spaces

TL;DR

This paper introduces a new metric combining pointed Gromov-Hausdorff and locally $C^0$ distances to analyze topological stability of maps between non-compact metric spaces, extending stability concepts.

Contribution

It develops a novel pointed $C^0$-Gromov-Hausdorff distance and applies it to define topologically $GH$-stable pointed homeomorphisms, bridging existing stability theories.

Findings

Defined a new pointed $C^0$-Gromov-Hausdorff distance.

Established the concept of topologically $GH$-stable pointed homeomorphisms.

Provided an example illustrating the impact of base point choice.

Abstract

We combine the pointed Gromov-Hausdorff metric [Ron10] with the locally $C^0$ distance to obtain the pointed $C^0$-Gromov-Hausdorff distance between maps of possibly different non-compact pointed metric spaces. The latter is then combined with Walters's locally topological stability [LNY18] and $GH$-stability from [AMR17] to obtain the notion of topologically $GH$-stable pointed homeomorphism. We give one example to show the difference between the distance when take different base point in a pointed metric space.