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arXiv · 2502.11615

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

Hiroki Nakajima·Takamitsu Yamauchi·Nicolò Zava
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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.

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Hiroki Nakajima, Takamitsu Yamauchi, Nicolò Zava. 2025-02-17. Topological dimension of the Gromov-Hausdorff and Gromov-Prokhorov spaces. https://arxiv.org/abs/2502.11615

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