Let $F : X → X$ be continuous. Prove that the set $\{x ∈ X : F(x) = x\}$ of fixed points of F is closed in X Here X is a Hausdorff Space.
I know that singleton sets, {x}, are closed in a Hausdorff space. Although Im not sure if thats how to use the Hausdorff property. 
Should I investigate $h=F(x)-x$?
Can anyone give a hint

I have this idea that I'd love somebody with more knowledge than I will check: define a function 
$$\;\phi:X\to X\times X\;,\;\;\text{by}\;\;\;\phi(x):=(x,F(x))\;$$
Since each coordinate function is continous also $\;\phi\;$ is, and if $\;\Delta:=\{(x,x)\in X\times X\}\;$ is the diagonal in the cartesian product, then
$$\phi^{-1}(\Delta\cap\phi(X))=\left\{x\in X\;;\;F(x)=x\;\right\}=:K$$
Since $\;\Delta\cap\phi(X)\;$ is closed  (see comments by Ayman Hourieh and Thomas Andres below the question) and $\;\phi\;$ is continous, then $\;K\;$ is closed.
