Continuous function from a compact space to a Hausdorff space is a closed function

Question

We have that $f\colon X\to Y$ is a continuous function, $X$ is a compact space and $Y$ is a Hausdorff space. Prove that $f$ is a closed function.

Answer 1

Let $C$ be a closed subset of $X$; you want to prove that $f(C)$ is closed in $Y$.

We use three basic facts about compact spaces:

1. a closed subset of a compact space is compact;
2. the image of a compact subset under a continuous function is compact;
3. a compact subset of a Hausdorff space is closed.

Now the proof of your statement.

Since $X$ is compact, $C$ is compact as well; therefore $f(C)$ is compact. A compact subset of a Hausdorff space is closed. Hence $f(C)$ is closed.