A complete Heyting algebra whose Scott space is non-sober

TL;DR

This paper constructs a complete Heyting algebra with a non-sober Scott space, answering a longstanding open problem, and also characterizes well-filtered spaces via their upper spaces.

Contribution

It demonstrates the existence of a complete Heyting algebra with a non-sober Scott space and characterizes well-filtered spaces through their upper spaces.

Findings

Existence of a complete Heyting algebra with non-sober Scott space.

The Scott space of $ ext{D}(L)$ is non-sober if that of $L$ is non-sober.

A $T_0$ space is well-filtered iff its upper space is well-filtered.

Abstract

We prove that (1) for any complete lattice $L$, the set $\mathcal{D}(L)$ of all nonempty 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 $\mathcal{D}(L)$ is non-sober. Using these results and the 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 give a positive answer to a problem posed by Jung. We will also prove that a $T_0$ space is well-filtered iff its upper space (the set $\mathcal{D}(X)$ of all nonempty saturated compact subsets of $X$ equipped with the upper Vietoris topology) is well-filtered, which answers another open problem.