The Scott Topology

In Chapter I we encountered the rich order theoretic structure of complete lattices and of continuous lattices. Wherever it was feasible to express statements on the level of generality of dcpo s and domains we did so. Perhaps even more typical for these partially ordered sets is their wealth of topological structure. The aim of the present chapter is to introduce topology into the study – a program to be continued in Chapter III. Section II-1 begins with a discussion of the Scott topology and its connection with the convergence given in order theoretic terms by lower limits, or liminfs. This leads to a characterization theorem for domains in terms of properties of their lattices of Scott open sets (II-1.14) – a type of theorem that will become a recurrent theme (see Chapter VII). One motivation for such considerations arises from the appearances of domain theory in theoretical computer science: one typically needs the generality of domains to model the structures and constructions under consideration, while continuous lattices enter the scene as their lattices of open sets. In Section II-2 we determine that the functions continuous for the Scott topology are those preserving directed sups. We can thus express one and the same property of a function between dcpo s either in topological or in order theoretical terms. The space [ S → T ] of all Scott-continuous functions between continuous lattices is itself a continuous lattice, and the category of continuous lattices proves to be cartesian closed .