It is not hard to show that if $f: X \rightarrow X$ is a continuous map and $X$ is a Hausdorff space, then the set of fixed points is closed in $X$. We basically just look at the diagonal and consider the map $g: X \rightarrow X \times X$ defined by $g(x)=(x,f(x))$.
What happens if we drop the condition that $X$ is Hausdorff? I guess the set of fixed points is not closed anymore. What would be an example? I tried looking at the cofinite topology but didn't find an example.