Two topologies on the lattice of Scott closed subsets

Abstract

For a poset P , let σ(P ) and Γ(P ) respectively denote the lattice of its Scott open subsets and Scott closed subsets ordered by inclusion, and set ΣP = (P, σ(P )). In this paper, we discuss the lower Vietoris topology and the Scott topology on Γ(P ) and give some sufficient conditions to make the two topologies equal. We built an adjunction between σ(P ) and σ(Γ(P )) and proved that ΣP is core-compact iff ΣΓ(P ) is core-compact iff ΣΓ(P ) is sober, locally compact and σ(Γ(P )) = υ(Γ(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.