I'm a little confused about certain assumptions usually made in GR as to how rigorous they are mathematically speaking. For instance the assumption generally presented without proof that GR spacetime manifold is a Hausdorff space seems not to be warranted given the fact that pseudometric spaces (with no definite positive metric) are not Hausdorff. Why make that assumption then? On the other hand the defining property of GR was explaining gravity thru curvature as an invariant, but Lorentzian manifolds, precisely due to their not being metric spaces, may be both flat and curved depending on what patch is chosen, in other words curvature is not a property of the manifold alone. Finally the assumption that the GR manifold is smooth seems to be contradicted by the existence of singularities, the condition usually imposed that one must only look at the space and time intervals that are singularity free doesn't seem a very rigorous mathematical prescription.