Let $X$ be a Hausdorff countably compact space and $Y$ first countably. If $f:X\to Y$ is a continuous bijection then it is a homeomorphism.
Like in the case of compact spaces, I'm trying to show $f$ is closed.
If $A\subseteq X$ is closed, $A$ es countably compact and so is $f(A)$. I want to define a countable open cover for $f(A)$ by using $Y$ is first countably, but I don't know how to do this.
Would you give me a hint?