Topologically speaking, a hole can be introduced into a manifold and it will still be a manifold, e.g. remove points within a 2-sphere of some radius from the cartesian plane and you'll still have a manifold.
Penrose's singularity theorems prove the existence (mathematically) of incomplete null geodesics, but not of curvature singularities, per say. So Im wondering if, in my naive view, maybe a topological perspective is better suited to describe curvature singularities? IF so, I'm not sure how to view curvature singularities topologically.
My question: is a curvature singularity, e.g. a black hole, in general relativity simply a topological hole of the spacetime manifold, or is it more topologically complicated? Or instead, is the singular structure not a part of the manifold, as suggested in @benrg's answer to this question, and thus is not a topological hole? Or is it not this simple, and there's some nuance(s) that I'm missing?
EDIT: I suppose I can phrase my confusion like this: how is it logically consistent to say that the physical singularity is not a part of the spacetime manifold (like how $\infty$ is not a point on the real line) AND that we can have a description of the singular structure from the metric itself (e.g., the Kerr metric has a ring singularity as can be shown from the metric)? Or do we avoid this confusion if the singularity is a topological hole?