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In [[mathematics]], given two [[partially ordered set]]s ''P'' and ''Q'', a [[Function (mathematics) | function]] <math>f : P \rightarrow Q</math> between them is '''Scott-continuous''' (named after the mathematician [[Dana Scott]]) if it [[limit preserving function (order theory)|preserves]] all directed suprema, i.e. if for every [[directed set|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'': {{nowrap begin}}sup ''f''(''D'') = ''f''(sup ''D'').{{nowrap end}}<ref name="Vickers1989">{{Cite book |last=Vickers |first=Steven |authorlink=Steve Vickers (academia) |title=Topology via Logic |publisher=[[Cambridge University Press]] |year=1989 |isbn=0-521-36062-5}}</ref>
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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 |authorlink1=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 show up in the study of models for [[lambda calculi]]<ref name=Scott1972></ref> and the [[denotational semantics]] of computer programs.
 
==Properties==
A Scott-continuous function is always [[monotone function|monotonic]].
 
A subset of a partially ordered set 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 topological space satisfying the T<sub>0</sub> separation axiom, 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 |series= |volume=Vol. III |year=1994 |publisher=Oxford University Press |isbn=0-19-853762-X |chapter=Domain theory |chapterurl=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 |author=Bauer, Andrej and Taylor, Paul |year=2009 |title=The Dedekind Reals in Abstract Stone Duality |journal=Mathematical Structures in Computer Science |volume=19 |pages=757–838 |publisher=[[Cambridge University Press]] |doi=10.1017/S0960129509007695 |url=http://PaulTaylor.EU/ASD/dedras/ |accessdate=October 8, 2010 }}</ref>
 
For '''CPO''', the [[cartesian closed category]] of [[complete partial order]]s, two particularly notable examples of Scott-continuous functions are [[currying|curry]] and [[apply]].<ref>{{cite book |last1=Barendregt |first1=H.P. |authorlink1=Henk Barendregt |title=The Lambda Calculus |year=1984 |publisher=North-Holland |isbn=0-444-87508-5}} ''(See theorems 1.2.13, 1.2.14)''</ref>
 
==See also==
* [[Alexandrov topology]]
* [[Upper topology]]
 
==Footnotes==
<references/>
 
==References==
* {{planetmath reference|id=9063|title=Scott Topology}}
 
[[Category:Order theory]]
[[Category:General topology]]

Latest revision as of 16:45, 6 January 2015

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