The open ball is a manifold, and the closed ball is a compact manifold-with-boundary which extends the open ball, in which the open ball is dense, in which all the new points are boundary points. Is it the only one? can every manifold be compactified with a boundary?
What if we allow other spaces, like a closed polyhedron? There is a morphism (of locally ringed spaces) from a closed convex polyhedron to a closed ball, which is a homeomorphism on the topological spaces. What other non-smooth compactifications-by-boundary can we expect to smooth-out like this?