Template:Short description 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.<ref name="Vickers1989">Template:Cite book</ref> 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.<ref>Template:Nlab</ref>
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.<ref name="Vickers1989"/>
The Scott topology was first defined by Dana Scott for complete lattices and later defined for arbitrary partially ordered sets.<ref name="Scott1972">Template:Cite book</ref>
Scott-continuous functions are used in the study of models for lambda calculi<ref name=Scott1972 /> and the denotational semantics of computer programs.
PropertiesEdit
A Scott-continuous function is always monotonic, meaning that if <math>A \le_{P} B</math> for <math>A, B \subset P</math>, then <math>f(A) \le_{Q} f(B)</math>.
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.<ref name="AbramskyJung1994"/>
A directed complete partial order (dcpo) with the Scott topology is always a Kolmogorov space (i.e., it satisfies the T0 separation axiom).<ref name="AbramskyJung1994"/> However, a dcpo with the Scott topology is a Hausdorff space if and only if the order is trivial.<ref name="AbramskyJung1994"/> The Scott-open sets form a complete lattice when ordered by inclusion.<ref name="BauerTaylor2009"/>
For any Kolmogorov space, the topology induces an order relation on that space, the specialization order: Template:Nowrap 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.<ref name="AbramskyJung1994">Template:Cite book</ref>
ExamplesEdit
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.<ref name="BauerTaylor2009">Template:Cite journal</ref>
Template:AnchorFor CPO, the cartesian closed category of dcpo's, two particularly notable examples of Scott-continuous functions are curry and apply.<ref>Template:Cite book (See theorems 1.2.13, 1.2.14)</ref>
Nuel Belnap used Scott continuity to extend logical connectives to a four-valued logic.<ref>N. Belnap (1975) "How Computers Should Think", pages 30 to 56 in Contemporary Aspects of Philosophy, Gilbert Ryle editor, Oriel Press Template:ISBN</ref>
See alsoEdit
FootnotesEdit
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