Two topologies on the lattice of Scott closed subsets

Abstract

For a poset $P$, let $\sigma(P)$ and $\Gamma(P)$ respectively denote the lattice of its Scott open subsets and Scott closed subsets ordered by inclusion, and set $\Sigma P=(P,\sigma(P))$. In this paper, we discuss the lower Vietoris topology and the Scott topology on $\Gamma(P)$ and give some sufficient conditions to make the two topologies equal. We built an adjunction between $\sigma(P)$ and $\sigma(\Gamma(P))$ and proved that $\Sigma P$ is core-compact iff $\Sigma\Gamma(P)$ is core-compact iff $\Sigma\Gamma(P)$ is sober, locally compact and $\sigma(\Gamma(P))=\upsilon(\Gamma(P))$ (the lower Vietoris topology). This answers a question in [17]. Brecht and Kawai [2] asked whether the consonance of a topological space $X$ implies the consonance of its lower powerspace, we give a partial answer to this question at the last part of this paper.