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

Abstract: We present several equivalent conditions of the continuity of the supremum function $\Sigma C(X)\times\Sigma C(X)\rightarrow\Sigma C(X)$ under mild assumptions, where $C(X)$ denotes the lattice of closed subsets of a $T_0$ topological space. We also provide an example of a non-monotone determined space $X$ such that $\eta=\lambda x.{\downarrow}x\colon X\rightarrow\Sigma C(X)$ is continuous. Additionally, we 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 a Scott completion. This allows us to give more examples which do not have Scott completions.