Fundamentals of Theory of Continuous Gromov--Hausdorff distance

TL;DR

This paper introduces a modified version of the Gromov--Hausdorff distance that incorporates topological differences, exploring its properties, distinctions from the classical version, and implications for metric space analysis.

Contribution

It defines the continuous GH-distance, analyzes its properties, and demonstrates how it differs from the classical GH-distance by accounting for topological features.

Findings

The continuous GH-distance shares many properties with the classical version.

It can differ significantly from the classical GH-distance when topologies differ.

The continuous GH-distance is intrinsic but incomplete.

Abstract

The Gromov--Hausdorff distance (hereinafter referred to as the GH-distance) is a measure of non-isometricity of metric spaces. In this paper, we study a modification of this distance that also takes topological differences into account. The resulting function of pairs of metric spaces is called the continuous GH-distance. We show that many basic properties of the classical GH-distance also hold in the continuous case. However, the continuous GH-distance, distinguishing between topologies, can differ significantly from the classical one. We will provide numerous examples of this distinction and demonstrate the role of topological dimension here. In particular, we will prove that the continuous GH-distance, like the classical one, is intrinsic, but, unlike the classical one, it is incomplete. Since we are dealing with all metric spaces, we will show, within the framework of the von Neumann-Bernays-G\"odel set theory, how topological concepts can be transferred to proper classes.