A co-sober Alexandrov space whose Scott power space is not co-sober

Abstract

We show that the Scott power construction does not preserve co-sobriety. For a dcpo $P$, let $\operatorname{Fin}(P)$ be the poset of nonempty finitely generated upper sets ordered by reverse inclusion. We prove that co-sobriety of $Σ\operatorname{Fin}(P)$ implies co-sobriety of $ΣP$. The proof adapts the point-saturation and union argument used for full compact saturated power posets; the finite step follows from a result of Heckmann and Keimel. Applying this transfer to a dcpo with a non-co-sober Scott space yields a co-sober Alexandrov space whose Scott power space is not co-sober. We also construct a countable example for which the Smyth power poset is countable. In the countable example, the same poset is co-sober under the upper Vietoris topology but not under the Scott topology.