Function spaces with compact-open topology need not be sober

This is the authors' abstract. We don't add key points for this paper.

We exhibit sober spaces X and Y such that the function space [X → 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 = pt(H) is sober with O(X) 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.