Can the Brouwer fixed point theorem be formulated for non-Hausdorff spaces ?
More particularly, is there a formulation of the Brouwer fixed point theorem which covers both the standard case of metrizable spaces, and the case of finite topological spaces ?
For example, is the following true:
Each endomorphism of a quasi-compact $T_0$ space which is an absolute extensor for normal spaces, necessarily has a fixed point.
As pointed out in comments by Michael Greinecker, Kinoshita, 1953 gives a counterexample to the following which is a contractible compact subset of $\Bbb R^3$, see also Bing,The elusive fixed point property and references in The generalization of Brouwer's fixed point theorem?.
Each endomorphism of a contractible quasi-compact $T_0$ space necessarily has a fixed point. (false!)
hence we replace "contractible" by "absolute extensor"
Note that being $T_0$ is necessary: the indiscrete space with two points is quasi-compact and contractible, yet the permutation of the two points has no fixed point. Perhaps one can relax the notion of a fixed point, e.g. by requiring that $x$ and $f(x)$ are topologically indistinguishable.
By the Brouwer theorem for finite topological spaces I mean what is implied by the Brouwer fixed point theorem for the geometric realisation of a finite topological space: namely, an endomorphism $f:K\to K$ of a finite topological space $K$ has a fixed point provided the endomorphism $\lvert f\rvert\colon\lvert K\rvert\to\lvert K\rvert$ of the geometric realisation of $K$ has a fixed point.
