I'm stuck on this statement. Could anyone please help me out?

Let $X$ be a compact manifold, every map $f: X \longrightarrow Y$ is proper.

The definition of proper: a smooth map between manifolds is called proper if inverse images of compact subsets are compact.

I know that continuous maps map compact sets to compact sets. But this seems to be the converse of that... Is there anything that I'm missing here? Thanks!


In $Y$, compact sets are closed (assuming $Y$ is Hausdorff). $f$ is continuous, so the inverse image of a closed set is closed. But a closed subset of a compact (Hausdorff) space is compact. So the inverse image of a compact set is compact.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.