Gromov's compactness theorem for the intrinsic timed-Hausdorff distance

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-Fréchet 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 Arzelà–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.