The Scott space of lattice of closed subsets with supremum operator as a topological semilattice

We present several equivalent conditions of the continuity of the supremum function from the square of the Scott space of $C(X)$ to itself under mild assumptions, where $C(X)$ denotes the lattice of closed subsets of a ${\mathbf{T}}_{\mathbf{0}}$ topological space. We also show that a ${\mathbf{T}}_{\mathbf{0}}$ space is quasicontinuous (quasialgebraic) iff the lattice of its closed subsets is a quasicontinuous (quasialgebraic) domain by using $n$-approximation. Furthermore, we provide a necessary condition for when a topological space possesses a Scott completion. This allows us to give more examples which do not have Scott completions.