In mathematics, given two partially ordered sets P and Q, a function f : P → Q between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema. That is, for every directed subset D of P with supremum in P, its image has a supremum in Q, and that supremum is the image of the supremum of D, i.e. <math>\sqcup f[D] = f(\sqcup D)</math>, where <math>\sqcup</math> is the directed join. When <math>Q</math> is the poset of truth values, i.e. Sierpiński space, then Scott-continuous functions are characteristic functions of open sets, and thus Sierpiński space is the classifying space for open sets.
A subset O of a partially ordered set P is called Scott-open if it is an upper set and if it is inaccessible by directed joins, i.e. if all directed sets D with supremum in O have non-empty intersection with O. The Scott-open subsets of a partially ordered set P form a topology on P, the Scott topology. A function between partially ordered sets is Scott-continuous if and only if it is continuous with respect to the Scott topology.
Scott-continuous functions are used in the study of models for lambda calculi
Examples
The open sets in a given topological space when ordered by inclusion form a lattice on which the Scott topology can be defined. A subset X of a topological space T is compact with respect to the topology on T (in the sense that every open cover of X contains a finite subcover of X) if and only if the set of open neighbourhoods of X is open with respect to the Scott topology.
For CPO, the cartesian closed category of dcpo's, two particularly notable examples of Scott-continuous functions are curry and apply.
Nuel Belnap used Scott continuity to extend logical connectives to a four-valued logic.
See also
- Alexandrov topology
- Upper topology