Problem 24.8.435C633451730

Problem 24.8.4.35C633451730 For a smooth big background $\theta $ on a manifold with bounded mass, theorem 13.27.4 gives a complete metrizable topology on the ordinary models of positive lower volume, in which the closure of the analytic models is exactly the set of ideal models. Extend this topology to ordinary models beyond bounded mass and positive lower volume, keeping the closure of the analytic models exactly the set of ideal models. The extension should keep the compatibility with smooth gauge changes and with two-sided perturbations of the background established in section 13.25, section 13.26, and it should treat limits of backgrounds that lose bigness or a uniform positive lower bound for the volume. On curves it should recover the complete topology of polar total variation of corollary 13.28.7 on all models. Determine whether the extension in higher dimension is completely metrizable.