It is known that for any locally compact Hausdorff space X, we can define a Hausdorff one-point compactification containing X. In the case of the (differentiable) manifold $\mathbb R^n$ this one-point compactification turns out to be (homeomorphic to) $\mathbb S^n$, which is again a (differentiable) manifold.
This leads to the following question:
What does the picture look like in the general case for compactifications of an arbitrary manifold $M$?
Although the one-point compactification of $M$ is not a manifold in general (e.g. $\mathbb R^n - 0$); is it possible to view every manifold as an open (dense?) subset of a compact manifold by taking some other kind of compactification? In the differentiable case? In the $C^0$-case?
I had thought along the following lines at first: By the Whitney embedding theorem, every manifold $M$ can be thought of as a closed submanifold of $\mathbb R^n$ for some $n$. And by embedding $\mathbb R^n$ into $\mathbb S^n$, we can think of $M$ as an embedded submanifold of a compact manifold. But I guess taking the closure of $M$ in $\mathbb S^n$ will not in general leave us with a manifold anymore (?), so this does not answer my question...
Has this been looked into?
Thanks for any thoughts.