This seems like such a basic question that it must have been answered before, but I can't seem to find an answer anywhere. The reason I'm asking this, is because in the study of the classification of manifolds, people study mainly closed manifolds (i.e. compact manifolds without boundary), and never seem to mention anything about the classification of open manifolds. This decision would make sense to me, however, if every open manifold were in fact an open submanifold of a closed manifold.
If, given some open manifold $M_0$, I try to construct a closed manifold $M$ containing $M_0$, all the approaches that would seem natural to me fail to give a manifold.
If we naively let $M=\overline{M_0}$ be the 1-point compactification, we could end up with $M$ being the figure 8, which is not a manifold. Even if $M_0$ is connected, then we could set $M_0=\mathbb{S}^2\setminus\{x,y\}$ and $\overline{M_0}$ would not be a manifold.
Is the problem here just that the 1-point compactification is too weak? Would this work with the Čech-Stone compactification, or would that fail too?
