From continuity to homeomorphism, compactness in domain

[jostpuur]

Is this claim true? Assume that $X,Y$ are topological spaces, and that all closed subsets of $X$ are compact. Then all continuous bijections $f:X\to Y$ are homeomorphisms.

It looks true on my notebook, but I don't have a reference, and I don't trust my skills. Just checking.

 

[quasar987]

There is this useful theorem that says "If X is compact and Y is Hausdorff, then any continuous bijection f:X-->Y is a homeo" and the proof goes like this:

to show: f(C) is closed for all C in X closed. Take such a C. Since X is compact, C is compact, so f(C) is compact. But compact sets in a Hausdroff and closed, QED.

Observe that to say that all closed subsets of X are compact is equivalent to saying that X is compact. So, modulo Y being Hausdorff, your claim is the above theorem.

 

[jostpuur]

I see. I must have used the Hausdorff assumption without noticing it.