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A celebrated theorem of Nagata and subsequent refinements to schemes and algebraic spaces say that over a not-completely-monstrous base scheme, any separated morphism can be openly immersed in a proper morphism.

The construction of such a compactification is involved and I have no big picture. For nice topological spaces, there are several constructions of compactifications: some small (Alexandroff), some large (Stone-Čech), and surely others I don't know of.

What sort of compactification is Nagata's compactification theorem about? Does it have a nice analogue for topological spaces? Topological manifolds? Smooth manifolds?

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The required condition for smooth manifolds (you can see it as an analogue of the finite type condition of Nagata's theorem) is that the 'ends' of the manifold aren't too complicated. The results of Browder-Levine-Livesay (https://www.jstor.org/stable/2373259) and others say that if M has f.g. homology and is 'connected at infinity', then it is the interior of a closed manifold with boundary. Otherwise you have obvious counterexamples like a surface of infinite genus.

Some googling turned up this thread, in a similar vein: Compactification theorem for differentiable manifolds ?

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