Let $X$ be a topological space. Define the *closed fixed point set property* to be that every continuous self-map $f:X\to X$ has fixed point set $\operatorname{Fix}(f)=\{x\in X\mid f(x)=x\}$ which is closed in $X$.

There is a cute result which says that Hausdorff (i.e. $T_2$) spaces satisfy this property. The proof is pretty straight forward (it is an exercise in a topology course I was teaching).

A space $X$ is $T_1$ if for all pairs $x, y\in X$ there exists a neighbourhood $N_x$ of $x$ that does not contain $y$. Hausdorff spaces satisfy this condition. I am currently considering a space which is $T_1$ but *not* Hausdorff, and would like to prove that this space satisfies the closed fixed point set property. So I'm looking for inspiration:

> What properties $P$ are there so that a $T_1$ space which additionally satisfies $P$
>- will satisfy the closed fixed point set property, and
>- will not necessarily be Hausdorff.