I am trying to establish some basic facts about spectral spaces. In relation to this I am looking for a proof of, or a counter example to, the statement that compact open subsets of a sober $T_1$ space are closed.
Recall that a topological space is sober if every closed irreducible subset has a generic point. Whence in a sober $T_1$ space the closed irreducible subsets are precisely the singletons.
Any help is appreciated.
You can find counterexamples in some fairly typical sober $T_1$ spaces that are not Hausdorff. For example, consider $\mathbb{N}\cup\{x,y\}$ where every point of $\mathbb{N}$ is isolated and the neighbourhoods of $x$ and $y$ are the cofinite subsets containing $x$ and $y$ respectively. Then $\mathbb{N}\cup\{x\}$ is compact open but not closed.