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.^[1] 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.^[2]

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.^[1]

The Scott topology was first defined by Dana Scott for complete lattices and later defined for arbitrary partially ordered sets.^[3]

Scott-continuous functions are used in the study of models for lambda calculi^[3] and the denotational semantics of computer programs.