# X is compact and Y is Hausdorff and connected prove a function is surjective

I need help proving the following.

A function $f:X\to Y$ is an open map if whenever $U$ is an open subset of $X$, then $f(U)$ is an open subset of $Y$. Let $X$ and $Y$ be topological spaces. prove that if $X$ is compact, $Y$ is Hausdorff and connected, and $f\colon X\rightarrow Y$ is a continuous open map, then $f$ is surjective.

Thank you!

Hint: $f(X)$ is open and closed in $Y$.
Recall that $Y$ is connected if and only if the only subsets of $Y$ which are both open and closed are $Y$ and $\emptyset$. Recall also that for $X$ compact and $Y$ Hausdorff, if $V$ is a closed subset of $X$, then $f(V)$ is a closed subset of $Y$ - that is, any continuous map $f\colon X\rightarrow Y$ is a closed map.
• $X$ is a closed and open subset of $X$. If $f$ is a closed and open map, then $f(X)$ is a closed and open subset of $Y$. (I can't say much more without writing the punchline of the proof) Oct 2, 2013 at 1:48