Scott continuity

2025-08-27T14:28:17Z

2600:4040:7D94:DB00:2252:CAEC:789C:F4E0: Type fixed (element, not subset)

{{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.

==Properties==
A Scott-continuous function is always [[monotone function|monotonic]], meaning that if <math>A \le_{P} B</math> for <math>A, B \in P</math>, then <math>f(A) \le_{Q} f(B)</math>.

A subset of a directed complete partial order is [[closed set|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|T<sub>0</sub> 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 (set theory)|inclusion]].<ref name="BauerTaylor2009"/>

For any Kolmogorov space, the topology induces an order relation on that space, the [[specialization order]]: {{nowrap|''x'' ≤ ''y''}} 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 space|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">{{cite book |last1=Abramsky |first1=S. |last2=Jung |first2=A. |editor1-first=S. |editor1-last=Abramsky |editor2-first=D.M. |editor2-last=Gabbay |editor3-first=T.S.E. |editor3-last=Maibaum |title=Handbook of Logic in Computer Science |volume=III |year=1994 |publisher=Oxford University Press |isbn=978-0-19-853762-5 |chapter=Domain theory |chapter-url=http://www.cs.bham.ac.uk/~axj/pub/papers/handy1.pdf }}</ref>

==Examples==
The open sets in a given topological space when ordered by [[inclusion (set theory)|inclusion]] form a [[lattice (order)|lattice]] on which the Scott topology can be defined. A subset ''X'' of a topological space ''T'' is [[compact space|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 neighbourhood]]s of ''X'' is open with respect to the Scott topology.<ref name="BauerTaylor2009">{{cite journal |author1=Bauer, Andrej |author2=Taylor, Paul |name-list-style=amp |year=2009 |title=The Dedekind Reals in Abstract Stone Duality |journal=Mathematical Structures in Computer Science |volume=19 |issue=4 |pages=757–838 |doi=10.1017/S0960129509007695 |url=http://PaulTaylor.EU/ASD/dedras/ |access-date=October 8, 2010 |citeseerx=10.1.1.424.6069 |s2cid=6774320 }}</ref>

{{anchor|curryApply}}For '''CPO''', the [[cartesian closed category]] of dcpo's, two particularly notable examples of Scott-continuous functions are [[currying|curry]] and [[apply]].<ref>{{cite book |last1=Barendregt |first1=H.P. |author-link1=Henk Barendregt |title=The Lambda Calculus |year=1984 |publisher=North-Holland |isbn=978-0-444-87508-2}} ''(See theorems 1.2.13, 1.2.14)''</ref>

[[Nuel Belnap]] used Scott continuity to extend [[logical connective]]s 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 {{ISBN|0-85362-161-6}}</ref>

==See also==
* [[Alexandrov topology]]
* [[Upper topology]]

==Footnotes==
<references/>

==References==
* {{planetmath reference|urlname=ScottTopology|title=Scott Topology}}

[[Category:Order theory]]
[[Category:General topology]]
[[Category:Domain theory]]

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Scott continuity