Quasicontinuous Domains and the Smyth Powerdomain

In Domain Theory quasicontinuous domains pop up from time to time generalizing slightly the powerful notion of a continuous domain. It is the aim of this paper to show that quasicontinuous domains occur in a natural way in relation to the powerdomains of finitely generated and compact saturated subsets. Properties of quasicontinuous domains seem to be best understood from that point of view. This is in contrast to the previous approaches where the properties of a quasicontinuous domain were compared primarily with the properties of the lattice of Scott-open subsets. We present a characterization of those domains that occur as domains of nonempty compact saturated subsets of a quasicontinuous domain. A set theoretical lemma due to M. E. Rudin has played a crucial role in the development of quasicontinuous domains. We present a topological variant of Rudin’s Lemma where irreducible sets replace directed sets. The notion of irreducibility here is that of a nonempty set that cannot be covered by two closed sets except if already one of the sets is covering it. Since directed sets are the irreducible sets for the Alexandroff topology on a partially ordered set, this is a natural generalization. It allows a remarkable characterization of sober spaces. For this we denote by QX the space of nonempty compact saturated subsets (with the upper Vietoris topology) of a topological space X. The following properties are equivalent: (1) X is sober, (2) QX is sober, (3) X is strongly well-filtered in the following sense: Whenever A is an irreducible subset of QX and U an open subset of X such that ⋂ A ⊆ U , then K ⊆ U for some K ∈ A. This result fills a gap in the existing literature.