On the Gromov-Hausdorff distance between the cloud of bounded metric spaces and a cloud with nontrivial stabilizer

TL;DR

This paper explores the structure of metric space classes called clouds under Gromov-Hausdorff distance, focusing on stabilizers and calculating distances between specific clouds like bounded spaces and the real line.

Contribution

It introduces the concept of clouds in the Gromov-Hausdorff framework and analyzes the stabilizer group, providing new insights into distances between these classes.

Findings

All clouds are proper classes.

The Gromov-Hausdorff distance between certain clouds is finite.

Calculated the distance between bounded metric spaces and the real line cloud.

Abstract

The paper studies the class of all metric spaces considered up to zero Gromov-Hausdorff distance between them. In this class, we examine clouds - classes of spaces situated at finite Gromov-Hausdorff distances from a reference space. We prove that all clouds are proper classes. The Gromov-Hausdorff distance is defined for clouds similarly with the case of that for metric spaces. A multiplicative group of transformations of clouds is defined which is called stabilizer. We show that under certain restrictions the distance between the cloud of bounded metric spaces and a cloud with a nontrivial stabilizer is finite. In particular, the distance between the cloud of bounded metric spaces and the cloud containing the real line is calculated.