The topology of Gromov--Hausdorff space

This paper is divided into four parts. We prove that the Gromov--Hausdorff space is homeomorphic to the Hilbert space. In Part I, we construct an assignment of a full-support probability measure to every nonempty compact metric space that respects isometries and is continuous for simultaneous Hausdorff convergence of the spaces and weak convergence of the measures. In Part II, we use these measures to construct finite-dimensional local models whose induced pseudometrics approximate the original distances uniformly and whose norms and point maps vary continuously up to orthogonal changes of coordinates. In Part III, we use the local models to prove that the Gromov--Hausdorff space is an absolute retract for all metrizable spaces. In Part IV, we establish a discrete approximation property and conclude that the space of isometry classes of nonempty compact metric spaces is homeomorphic to the real separable infinite-dimensional Hilbert space.