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Let $X$ be a countably compact space and $Y$ a second-countable, Hausdorff space. Show that if $f:X\rightarrow Y$ is a bijective continuous function then it is a homeomorphism.

Is easy to see that $Y$ is a countably compact space and then a compact metric space (by the urysohn's metrization theorem), and also we can conclude that $X$ is a hausdorff, normal space. but I can't do much more, can anyone help me?

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stefan is correct! You basically need the following facts, 1. the closed subspace of countably compact space is countably compact. 2. the continuous image of a countably compact space is countably compact 3. second countable space is Lindolef (any open cover has a countable subcover). Then you can use the same method as showing "compact in Hausdorff closed" to show.

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Take a closed set $A\subseteq X$. Then $A$ is countably compact, and so is $B=f(A)$. We'll show that $f(A)$ is closed. Let $y\notin f(A)$. For each $b\in B$ there are disjoint basic open sets $U_b$ around $y$ and $V_b$ around $b$. Now use that $B$ is countably compact ...

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