A direct characterization of the monotone convergence space completion

For a T0-space X and the set Γ(X) of all closed subsets of X, the d-closure of all point closures of X in Γ(X) endowed with the lower Vietoris topology is called the standard D-completion of X. In this paper, by introducing the notion of a tapered set, we present a direct characterization of the D-completion: the set of all tapered closed subsets of X endowed with the lower Vietoris topology is exactly the standard D-completion of X.