A complete Heyting algebra whose Scott space is non-sober
Tom 252 / 2021
Streszczenie
We prove that (1) for any complete lattice $L$, the set $\mathcal {D}(L)$ 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 $\mathcal {D}(L)$ 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 $T_0$ space is well-filtered iff its upper space (the set $\mathcal {D}(X)$ of all non-empty saturated compact subsets of $X$ equipped with the upper Vietoris topology) is well-filtered, which answers another open problem.
