Two topologies on the lattice of Scott closed subsets [article]

Yu Chen, Hui Kou, Zhenchao Lyu
2021 arXiv   pre-print
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 Σ P=(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 Σ P 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
more » ... 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.
arXiv:2103.15139v1 fatcat:gyuudym57bdtrhpo4235cvw5ky