Scott continuity

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. ${\displaystyle \sqcup f[D]=f(\sqcup D)}$, where ${\displaystyle \sqcup }$ is the directed join. When ${\displaystyle Q}$ 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.

The Scott topology was first defined by Dana Scott for complete lattices and later defined for arbitrary partially ordered sets.

Scott-continuous functions are used in the study of models for lambda calculi and the denotational semantics of computer programs.