2
$\begingroup$

Let X be a Hausdorff k-space (Hausdorff compactly generated space) and h a bijection on X such that for any subset E of X we have

E compact <=> h(E) compact. 

Question: Does it follow that h is a homeomorphism? (The converse is true for any space X since the continuous image of a compactum is compact).

Background: I am trying to see if it is possible to define homeomorphisms on Hausdorff k-spaces solely in terms of preservation of compact sets. More generally, I would like to identify topological spaces for which such a characterization of homeomorphisms would be possible.

Thanks, Pouya

$\endgroup$
0

1 Answer 1

4
$\begingroup$

Yes. If you restrict $h$ to any compact subset $E$, then $h$ gives a homeomorphism from $E$ to $h(E)$, because a subset of $E$ (or $h(E)$) is closed iff it is compact, so $h$ and its inverse both preserve closed sets. By compact generation, this implies that both $h$ and its inverse are continuous, so $h$ is a homeomorphism on the whole space.

$\endgroup$
1
  • 1
    $\begingroup$ Thanks, in the meanwhile I made a very similar reasoning, which I post below (partly for my own reference; I tend to lose my notes). Take any compactum $K \subset X$. Then $h(K)$ is compact by assumption. Thus any closed set $C\subset h(K)$ is also compact, being a closed subset of a compactum. Therefore $h^{-1}(C)$ is compact, and hence closed ($X$ is Hausdorff). Thus the pre-image of any closed set $C$ under $h|_K$ is closed, i.e. $h|_K$ is continuous. By compact generation, $h$ itself is continuous. Similarly, one shows that $h^{-1}$ is continuous, i.e. $h$ is a homeomorphism. $\endgroup$
    – pdt
    Mar 16, 2011 at 16:00

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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