$X$ Hausdorff and Compact, $f: X \rightarrow X$ continuous. Is the set of fixed points of $f$ Compact?

Question

I'm trying to prove the following statement:

If $X$ is a Compact, Hausdorff Topological Space and $f:X\rightarrow X$ is a continuous function, then the set $F=\left\{ x \in X : f(x) = x\right\}$ of fixed points of the function $f$ is Compact.

Yet any clues on how to even start.

Answer 1

The set of fixed points is closed in $X$. This is because it is the inverse image of the diagonal $\Delta=\{(x,x):x\in X\}\subseteq X\times X$ under the continuous map $x\mapsto (x,f(x))$ from $X$ to $X\times X$. Note that $\Delta$ is closed in $X\times X$ due to the Hausdorff property.

As the fixed points are a closed subset of a compact space, they also form a compact space.