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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 lowerarXiv:2103.15139v1 fatcat:gyuudym57bdtrhpo4235cvw5ky