Scott continuity: Difference between revisions
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{{Short description|Definition of continuity for functions between posets}} |
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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 [[limit preserving function (order theory)|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> |
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 [[limit preserving function (order theory)|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> |
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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> |
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> |
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Scott-continuous functions |
Scott-continuous functions are used in the study of models for [[lambda calculi]]<ref name=Scott1972 /> and the [[denotational semantics]] of computer programs. |
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==Properties== |
==Properties== |
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A Scott-continuous function is always [[monotone function|monotonic]]. |
A Scott-continuous function is always [[monotone function|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>. |
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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 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"/> |
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Latest revision as of 09:40, 6 January 2024
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. , where is the directed join.[1] When 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.[2]
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.[3]
Scott-continuous functions are used in the study of models for lambda calculi[3] and the denotational semantics of computer programs.
Properties
[edit]A Scott-continuous function is always monotonic, meaning that if for , then .
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.[4]
A directed complete partial order (dcpo) with the Scott topology is always a Kolmogorov space (i.e., it satisfies the T0 separation axiom).[4] However, a dcpo with the Scott topology is a Hausdorff space if and only if the order is trivial.[4] The Scott-open sets form a complete lattice when ordered by inclusion.[5]
For any Kolmogorov space, the topology induces an order relation on that space, the specialization order: 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: 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.[4]
Examples
[edit]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.[5]
For CPO, the cartesian closed category of dcpo's, two particularly notable examples of Scott-continuous functions are curry and apply.[6]
Nuel Belnap used Scott continuity to extend logical connectives to a four-valued logic.[7]
See also
[edit]Footnotes
[edit]- ^ a b Vickers, Steven (1989). Topology via Logic. Cambridge University Press. ISBN 978-0-521-36062-3.
- ^ Scott topology at the nLab
- ^ a b Scott, Dana (1972). "Continuous lattices". In Lawvere, Bill (ed.). Toposes, Algebraic Geometry and Logic. Lecture Notes in Mathematics. Vol. 274. Springer-Verlag.
- ^ a b c d Abramsky, S.; Jung, A. (1994). "Domain theory" (PDF). In Abramsky, S.; Gabbay, D.M.; Maibaum, T.S.E. (eds.). Handbook of Logic in Computer Science. Vol. III. Oxford University Press. ISBN 978-0-19-853762-5.
- ^ a b Bauer, Andrej & Taylor, Paul (2009). "The Dedekind Reals in Abstract Stone Duality". Mathematical Structures in Computer Science. 19 (4): 757–838. CiteSeerX 10.1.1.424.6069. doi:10.1017/S0960129509007695. S2CID 6774320. Retrieved October 8, 2010.
- ^ Barendregt, H.P. (1984). The Lambda Calculus. North-Holland. ISBN 978-0-444-87508-2. (See theorems 1.2.13, 1.2.14)
- ^ 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