Gromov's Compactness Theorem for the Intrinsic Timed-Hausdorff Distance

Abstract

The intrinsic timed-Hausdorff distance between timed-metric spaces, first introduced by Sakovich--Sormani, yields a weak notion of convergence for space-times. In this paper we prove a compactness theorem for the intrinsic timed-Hausdorff convergence of timed-metric spaces using timed-Fr\'echet maps. Our proof introduces the notion of "addresses" and provides a new way of stating Gromov's original compactness theorem for Gromov--Hausdorff convergence of metric spaces. We also obtain a new Arzel\`a--Ascoli theorem for real valued uniformly bounded Lipschitz functions on Gromov--Hausdorff converging compact metric spaces. Moreover, we establish the triangle inequality for the intrinsic timed-Hausdorff distance.