The topology of Gromov--Hausdorff space

Abstract

We prove that the Gromov--Hausdorff space is homeomorphic to the real Hilbert space ${\ell }^2$. We construct a continuous assignment of full-support probability measures that is equivariant under isometries and finite-dimensional local approximations that control all pairwise distances. These approximations yield the absolute retract property for all metrizable spaces. We also prove that any countable family of continuous maps from compact metrizable spaces can be approximated, with respect to a prescribed open cover, by maps whose images form a discrete family.

arXiv comment118 pages, 10 figures