D-completions and the d-topology

In this article we study a type of directed completion for posets and T0-spaces introduced by O. Wyler [Wy81] that we call the D-completion. The category D of monotone convergence spaces is a full reflective subcategory of the category of topological spaces and continuous maps, and the D-completion gives the reflection of a topological space into D, and as such exhibits a useful universal property that characterizes it. The direct definition and theory of the D-completion is based on a variant of the Scott topology, which we call the d-topology. For partially ordered sets the D-completion turns out to be a natural dcpo-completion that generalizes the rounded ideal completion.