  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.
