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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

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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.

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    $\begingroup$ 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
    Commented Mar 16, 2011 at 16:00

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