Editing Scott continuity (section)

{{Short description|Definition of continuity for functions between posets}}
In [[mathematics]], given two [[partially ordered set]]s ''P'' and ''Q'', a [[Function (mathematics)|function]] ''f'': ''P'' → ''Q'' between them is '''Scott-continuous''' (named after the mathematician [[Dana Scott]]) if it preserves all [[directed supremum|directed suprema]]. That is, for every [[directed subset]] ''D'' of ''P'' with [[supremum]] in ''P'', its [[image (mathematics)|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">{{Cite book |last=Vickers |first=Steven |author-link=Steve Vickers (academia) |title=Topology via Logic |publisher=[[Cambridge University Press]] |year=1989 |isbn=978-0-521-36062-3}}</ref> When <math>Q</math> is the poset of truth values, i.e. [[Sierpiński space]], then Scott-continuous functions are [[Indicator function|characteristic functions]] of open sets, and thus Sierpiński space is the classifying space for open sets.<ref>{{nlab|id=Scott+topology|title=Scott topology}}</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 (set theory)|intersection]] with ''O''. The Scott-open subsets of a partially ordered set ''P'' form a [[topological space|topology]] on ''P'', the '''Scott topology'''. A function between partially ordered sets is Scott-continuous if and only if it is [[continuous function (topology)|continuous]] with respect to the Scott topology.<ref name="Vickers1989"/>

The Scott topology was first defined by Dana Scott for [[complete lattice]]s and later defined for arbitrary partially ordered sets.<ref name="Scott1972">{{cite book |last1=Scott |first1=Dana |author-link1=Dana Scott |editor1-last=Lawvere |editor1-first=Bill |editor1-link=Bill Lawvere |title=Toposes, Algebraic Geometry and Logic |series=Lecture Notes in Mathematics |volume=274 |year=1972 |publisher=Springer-Verlag |chapter=Continuous lattices}}</ref>

Scott-continuous functions are used in the study of models for [[lambda calculi]]<ref name=Scott1972 /> and the [[denotational semantics]] of computer programs.