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

Abstract: We prove Gromov's Compactness Theorem for the Intrinsic timed Hausdorff ($\tau-H$) 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 (GH) convergence of metric spaces. We also obtain a new Arzela-Ascoli Theorem for real valued uniformly bounded Lipschitz functions on GH converging compact metric spaces. The intrinsic timed-Hausdorff distance between timed-metric-spaces was first defined by Sakovich-Sormani to define a weak notion of convergence for space-times, and our compactness theorem will soon be applied to advance their work in this direction.