# Is there a continuous 1-1 function from a compact Hausdorff space X onto a compact Hausdorff space Y?

Let f be a one-to-one function from a compact Hausdorff space X onto a compact Hausdorff space Y. Show that if f[K] is compact for every compact K subset of X, then f is continuous.

I know that f restricted to K must be continuous, because the continuous image of a compact set is compact. I'm not sure how to prove that f is continuous, other than the fact that every b in Y has a distinct f(b) in X because f is one-to-one and onto.

I'm not sure how to set this problem up.

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"I know that f restricted to K must be continuous, because the continuous image of a compact set is compact." You're mixing up your implication; consider any surjective map from [0,1] to {0,1}. – Gaffney Apr 26 '14 at 19:51

Hint: show that $f$ is a closed map. Then $f^{-1}$ is continuous, so $f^{-1}(K)$ is compact when $K$ is. Use this to show that $f$ is continuous.
@user114634: To show $f$ is a closed map, you have to start with an arbitrary closed subset $A\subseteq X$ and show that $f(A)$ is closed in $Y$. – Jack Lee Apr 26 '14 at 22:05
@user114634 A map is continuous iff the inverse image of every closed set is closed. Since $f$ is a bijective closed map, $f^{-1}$ is continuous. Now use the fact that continuous maps preserve compact sets ($f^{-1}$ in particular) to show that $f$ is also a closed map. – Seth Apr 26 '14 at 23:58