Function spaces with compact-open topology need not be sober

We exhibit sober spaces $X$ and $Y$ such that the function space $[X \to Y]_K$ of continuous maps, equipped with the compact-open topology, is not sober; the codomain can be chosen to be the Sierpiński space $S$. Our construction builds on Isbell's complete lattice $M$, whose Scott space $ΣM$ is non-sober. Let $H = Q(M)$ be the set of nonempty compact saturated subsets of $ΣM$, ordered by reverse inclusion. Since $H$ is well known to be a spatial frame, the space $X = \mathrm{pt}(H)$ is sober with $\mathcal{O}(X) \cong H$. The key technical result is that sobriety of the space $H$ would entail sobriety of the original Scott space $ΣM$. This contradicts Isbell's example, and the desired counterexample follows.