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



homeomorphisms on k-spaces
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.


