Question 8-8 in Lee's Introduction to smooth manifolds asks us to show that if $M \subset N$ is an embedded submanifold then it is closed iff the inclusion map is proper. Equivalently, a smooth embedding $g:M \to N$ is proper iff its image is closed.

Do we really need embedding for this -or even the smoothness? It seems to me, that the only relevant hypothesis is that the topologies on $M,N$ are metrizable, and that $g$ is a homeomorphism onto its image.

Any ideas?

  • 3
    $\begingroup$ You're right -- in fact, it's true even more generally than that. The second edition of my book Introduction to Topological Manifolds shows (pp. 118-121) that for topological spaces $M$ and $N$, a topological embedding $g:M\to N$ is proper iff its image is closed, provided that $N$ is a compactly generated Hausdorff space (which includes, in particular, all metrizable spaces and all first-countable Hausdorff spaces). $\endgroup$ – Jack Lee Oct 17 '12 at 19:05

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