Stability in Infinite Dimensions

Determine whether a general topological stability theorem holds in infinite-dimensional metric spaces; namely, whether sequences of compact infinite-dimensional metric spaces that converge to a limit space exhibit stabilization of topology in the tail, so that all sufficiently large terms are mutually homeomorphic to the limit space.

Background

Classical finiteness and stability results in Riemannian and metric geometry—such as those by Cheeger, Grove–Petersen–Wu, and Perelman's stability theorem for Alexandrov spaces—concern finite-dimensional spaces and leverage Gromov–Hausdorff convergence to deduce topological stabilization of convergent sequences.

The paper highlights that, prior to their work, such stability considerations had not been established in infinite-dimensional settings. They then focus on Wasserstein spaces—an important class of infinite-dimensional metric spaces—and prove stability and finiteness results in that context, leaving open the broader question of stability across general infinite-dimensional spaces.