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. $\sqcup f[D]=f(\sqcup D)$ , where $\sqcup$ is the directed join.^[1] When $Q$ 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.

^ ^a ^b Vickers, Steven (1989). Topology via Logic. Cambridge University Press. ISBN 978-0-521-36062-3.
^ Scott topology at the nLab
^ ^a ^b Scott, Dana (1972). "Continuous lattices". In Lawvere, Bill (ed.). Toposes, Algebraic Geometry and Logic. Lecture Notes in Mathematics. Vol. 274. Springer-Verlag.

[Vickers1989-1] Vickers, Steven (1989). Topology via Logic. Cambridge University Press. ISBN 978-0-521-36062-3.

[2] Scott topology at the nLab

[Scott1972-3] Scott, Dana (1972). "Continuous lattices". In Lawvere, Bill (ed.). Toposes, Algebraic Geometry and Logic. Lecture Notes in Mathematics. Vol. 274. Springer-Verlag.

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