A note on duals of topologies

In this note it is proved that for a quasicontinuous lattice L  , the lower topology ω(L)ω(L) and the Scott topology σ(L)σ(L) are duals for each other; and if L   is a complete lattice such that σ(L)σ(L) is continuous but not hypercontinuous (equivalently, L   is not quasicontinuous), then ω(L)ω(L) is not the dual of σ(L)σ(L) and hence they are not duals for each other.