D-completions and the d-topology
In this article we give a general categorical construction via reflection functors for various completions of T0-spaces subordinate to sobrification, with a particular emphasis on what we call the D-completion, a type of directed completion introduced by Wyler [O. Wyler, Dedekind complete posets and Scott topologies, in: B. Banaschewski, R.-E. Hoffmann (Eds.), Continuous Lattices Proceedings, Bremen 1979, in: Lecture Notes in Mathematics, vol. 871, Springer Verlag, 1981, pp. 384-389]. A key result is that all completions of a certain type are universal, hence unique (up to homeomorphism). We give a direct definition of the D-completion and develop its theory by introducing 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. In the latter part of the paper we consider settings in which the D-completion agrees with the sobrification respectively the closed ideal completion. © 2008 Elsevier B.V. All rights reserved.
Publication Source (Journal or Book title)
Annals of Pure and Applied Logic
Keimel, K., & Lawson, J. (2009). D-completions and the d-topology. Annals of Pure and Applied Logic, 159 (3), 292-306. https://doi.org/10.1016/j.apal.2008.06.019