Topological dimension of the Gromov-Hausdorff and Gromov-Prokhorov spaces

TL;DR

This paper investigates the topological dimension of Gromov-Hausdorff and Gromov-Prokhorov spaces, revealing finite and infinite-dimensional properties depending on the size of the spaces considered.

Contribution

It provides explicit calculations of the topological dimension for spaces of finite metric and metric measure spaces, highlighting their strong countable or infinite dimensionality.

Findings

Dimension of finite metric space classes is rac{n(n-1)}{2}.

Dimension of finite metric measure space classes is rac{(n+2)(n-1)}{2}.

Spaces of all such classes are strongly infinite-dimensional when sizes are unbounded.

Abstract

The Gromov-Hausdorff distance is a dissimilarity metric capturing how far two spaces are from being isometric. The Gromov-Prokhorov distance is a similar notion for metric measure spaces. In this paper, we study the topological dimension of the Gromov-Hausdorff and Gromov-Prokhorov spaces. We show that the dimension of the space of isometry classes of metric spaces with at most $n$ points endowed with the Gromov-Hausdorff distance is $\frac{n(n-1)}{2}$, and that of mm-isomorphism classes of metric measure spaces whose support consists of $n$ points is $\frac{(n+2)(n-1)}{2}$. Hence, the spaces of all isometry classes of finite metric spaces and of all mm-isomorphism classes of finite metric measure spaces are strongly countable dimensional. If, instead, the cardinalities are not limited, the spaces are strongly infinite-dimensional.