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.
\sqcupf[D]=f(\sqcupD)
\sqcup
Q
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.[2]
Scott-continuous functions are used in the study of models for lambda calculi and the denotational semantics of computer programs.
A Scott-continuous function is always monotonic, meaning that if
A\lePB
A,B\subsetP
f(A)\leQf(B)
A subset of a directed complete partial order is closed with respect to the Scott topology induced by the partial order if and only if it is a lower set and closed under suprema of directed subsets.
A directed complete partial order (dcpo) with the Scott topology is always a Kolmogorov space (i.e., it satisfies the T0 separation axiom). However, a dcpo with the Scott topology is a Hausdorff space if and only if the order is trivial. The Scott-open sets form a complete lattice when ordered by inclusion.
For any Kolmogorov space, the topology induces an order relation on that space, the specialization order: if and only if every open neighbourhood of x is also an open neighbourhood of y. The order relation of a dcpo D can be reconstructed from the Scott-open sets as the specialization order induced by the Scott topology. However, a dcpo equipped with the Scott topology need not be sober: the specialization order induced by the topology of a sober space makes that space into a dcpo, but the Scott topology derived from this order is finer than the original topology.[3]
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.[4]
For CPO, the cartesian closed category of dcpo's, two particularly notable examples of Scott-continuous functions are curry and apply.[5]
Nuel Belnap used Scott continuity to extend logical connectives to a four-valued logic.[6]