I read the result that every compact $n$-manifold is a compactification of $\mathbb{R}^n$.
Now, for surfaces, this seems clear: we take an n-gon, whose interior (i.e., everything in the n-gon except for the edges) is homeo. to $\mathbb{R}^n$, and then we identify the edges to end up with a surface that is closed and bounded.
We can do something similar with the $S^n$'s ; by using a "1-gon" (an n-disk), and identifying the boundary to a point. Or we can just use the stereo projection to show that $S^n-\{{\rm pt}\}\sim\mathbb{R}^n$; $S^n$ being compact (as the Alexandroff 1-pt. -compactification of $\mathbb{R}^n$, i.e., usual open sets + complements of compact subsets of $\mathbb{R}^n$). And then some messy work helps us show that $\mathbb{R}^n$ is densely embedded in $S^n$.
But I don't see how we can generalize this statement for any compact n-manifold. Can someone suggest any ideas?
Thanks in Advance.