Just parallelling this question, that seemed not to admit an easy answer at all, let's "soft down" the category and ask the same thing in the case of $\mathcal{C}^{\infty}$-differentiable manifolds [Edit: we consider only manifolds without boundary].

Well, so:

Is every differentiable manifold diffeomorphic to an open submanifold of a compact one?

Edit: As some comments have pointed out, there are manifolds for which the compactification theorem fails, so someone has suggested to change the question to the more meaningful:

Which differentiable manifolds are diffeomorphic to an open submanifold of a compact one?

Just parallelling this question, that seemed not to admit an easy answer at all, let's "soft down" the category and ask the same thing in the case of $\mathcal{C}^{\infty}$-differentiable manifolds.

Well, so:

Is every differentiable manifold diffeomorphic to an open submanifold of a compact one?