0
$\begingroup$

Let $X$ and $Y$ be Hausdorff spaces and suppose that $Y$ is locally compact. Let $f:X\to Y$ be a surjective map such that for any compact subset $K \subset Y$ the pre-image $$f^{-1}(K)=\{x\in X: f(x)\in K\}$$ is a compact subset of $X$. What can I tell about the continuity of $f$?

If $Y$ is compact, then $f$ is certainly continuous if we restrict $f$ to the pre-image of a compact, that is, $f|_{f^{-1} (K)}$, then it is continuous. Can I conclude anything else?

$\endgroup$
  • $\begingroup$ I don't think you will get continuity, because the condition you give is included in the definition of a proper map (together with continuity) $\endgroup$ – erz Aug 30 '18 at 0:52
2
$\begingroup$

$f$ need not be continuous. Let $Y$ be a countably infinite set with the discrete topology. Let $X$ be the same set with a different Hausdorff topology. Let $f$ be the identity map.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

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