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For a poset $P$, let $\sigma(P)$ and $\Gamma(P)$ respectively denote thelattice of its Scott open subsets and Scott closed subsets ordered byinclusion, and set $\Sigma P=(P,\sigma(P))$. In this paper, we discuss thelower Vietoris topology and the Scott topology on $\Gamma(P)$ and give somesufficient conditions to make the two topologies equal. We built an adjunctionbetween $\sigma(P)$ and $\sigma(\Gamma(P))$ and proved that $\Sigma P$ iscore-compact iff $\Sigma\Gamma(P)$ is core-compact iff $\Sigma\Gamma(P)$ issober, locally compact and $\sigma(\Gamma(P))=\upsilon(\Gamma(P))$ (the lowerVietoris topology). This answers a question in [17]. Brecht and Kawai [2] askedwhether the consonance of a topological space $X$ implies the consonance of itslower powerspace, we give a partial answer to this question at the last part ofthis paper.

Two topologies on the lattice of Scott closed subsets