Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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). – Jack Lee Oct 17 '12 at 19:05
Thank you a lot! – Bernard Oct 18 '12 at 4:01

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.