A complete Heyting algebra whose Scott space is non-sober

Abstract

We prove that (1) for any complete lattice 𝓛, the set 𝒟 (𝓛) of all non-empty saturated compact subsets of the Scott space of 𝓛 is a complete Heyting algebra (with the reverse inclusion order); and (2) if the Scott space of a complete lattice 𝓛 is non-sober, then the Scott space of 𝒟 (𝓛) is non-sober. Using these results and Isbell's example of a non-sober complete lattice, we deduce that there is a complete Heyting algebra whose Scott space is non-sober, thus giving an affirmative answer to a problem posed by Achim Jung. We also prove that 𝚊 𝒯₀ space is well-filtered iff its upper space (the set 𝒟 (𝓧) of all non-empty saturated compact subsets of X equipped with the upper Vietoris topology) is well-filtered, which answers another open problem.