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.