Infinite sober space

Question

Do there exists any concrete examples of infinite topological spaces that are sober but not Hausdorff? (The only example I have of a sober space that is non-Hausdorff is the Sierpinski space and variations thereof, which are finite.)

Answer 1

Certainly. Here's probably the simplest example. Let $S$ be any infinite set and let $X=S\cup\{g\}$, where $g\not\in S$. Take the topology on $X$ where the nonempty open sets are the cofinite sets that contain $g$. This is sober: every point except $g$ is closed, but the closure of $g$ is all of $X$, which is the only irreducible closed set with more than one point.

Or, you could note that sober spaces are closed under arbitrary disjoint unions and arbitrary products. So, you could get an infinite non-Hausdorff sober space by taking an infinite disjoint union or infinite product of finite non-Hausdorff sober spaces.

More generally, lots of natural examples come from algebraic geometry: for any commutative ring $R$, the space $\operatorname{Spec} R$ of prime ideals in $R$ is sober (and is not Hausdorff unless $R$ is $0$-dimensional). The first example above can be obtained by letting $R=k[x]$ for some field $k$, for instance.