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

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 subspaces 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 $n(n-1)/2$, and that of mm-isomorphism classes of metric measure spaces whose support consists of $n$ points endowed with the Gromov–Prokhorov distance is $(n+2)(n-1)/2$. 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.