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

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.