This is a follow up to one of my previous questions (81714). Suppose that $M$ is an open manifold, say with a single end. Previously, I was concerned with realizing $M$ as the interior of a compact manifold with boundary $\bar{M}$, essentially adding a boundary to $M$. The responses there were quite helpful, but even after some more reading I hesitate to say that I have a satisfactory understanding of the situation.

Now I would like to consider a similar question in a slightly different direction. I wonder how to realize $M$ as a dense open subset of a compact manifold $\bar{M}$, now *without* boundary. That is, the extra points $C=\bar{M}\setminus M$ are all *interior* points of $\bar{M}$. My question is two-fold.

a. Given $M$, what are the possibilities for $C$ and $M$ as topological spaces/manifolds? Is there an analog equivalence relation between possible choices for $C$ like $h$-cobordism for potential boundaries (as in my previous question)?

b. If $C$ and $M$ are somehow fixed in advance (one example of how, though not the exact one I'm most interested in, is to consider $C$ to be a set of points supplied by Cauchy completion with respect to some metric on $M$), What do I need to give $\bar{M}=M\cup C$ manifold structure? In the previous case of adding $C$ as a boundary, I would need a collar neighborhood of the end of $M$ of the form $C\times[0,1)$. But it is not clear to me what I would need to give $C$ the structure of a set of interior points of $\bar{M}$.

Again, I would be interested in answers both in the smooth and topological categories. Specific pointers to the literature are also appreciated.