TITLE: Does a Space with the Finite Closed Topology have Fixed Point Property?
QUESTION [0 upvotes]: I wanted to know if a space $(X, \tau)$ with $\tau$ the finite-closed topology, have the Fixed Point Property, that is if every continuous mapping $f$ of $(X, \tau)$ into itself has a fixed point.

REPLY [3 votes]: The topology you have in mind - more commonly called the cofinite topology - looks a lot like the discrete topology in a few ways. One of these ways is the following:

Every self-bijection is continuous with respect to the cofinite topology.

The proof is basically immediate: the preimage of a cofinite set under an injective map must be cofinite.
Now if $X$ is any set with more than one element, there is a permutation of $X$ with no fixed points (exercise$^1$), so in fact no space with the cofinite topology which has more than one point has the fixed point property.

$^1$Amusingly, this actually requires a small amount of the axiom of choice: it is consistent with $\mathsf{ZF}$ (= set theory without choice) that there is a set with more than one element such that every self-bijection of the set has a fixed point. In fact it is consistent with $\mathsf{ZF}$ that there is an infinite set $A$ such that every self-bijection of $A$ fixes all but finitely many points! This is very difficult to prove however (as all $\mathsf{ZF}$-consistency results are), but Cohen's original model of $\mathsf{ZF+\neg AC}$ provides an example. In such a model we do indeed have infinite sets which, when equipped with the cofinite topology, have the fixed point property. But these are extremely pathological models.
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