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 𝐓_0 topological space. We also show that a 𝐓_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 ...