Is a gravitational singularity distinct from a topological hole?

Question

In the context of a spacetime (e.g., definition 1 here) of classical general relativity theory, what is the technical mathematical difference(s) between a gravitational singularity and a topological hole? The two seem similar, naively, in the sense that neither are considered to be "part" of the manifold, and that one cannot continuously shrink the manifold beyond it.

Answer 1

If you take a geodesically complete manifold and remove a disk (make a hole) where at least one geodesic passes through, then you get a manifold with a hole which is geodesically incomplete and you have that sort of spacetime singularity.

On the other hand, a singularity of spacetime is not necessarily a hole. For instance, consider the maximal extension of Schwarzschild spacetime, which is homeomorphic to a contractible region in $R^4$. The region $r=0$ where the singularity lies is not topologically a hole but a spacelike hypersurface $T^2-X^2=1$. Here $T$ and $X$ are the Kruskal-Szekeres coordinates.