Scott continuity
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]\displaystyle{ \sqcup f[D] = f(\sqcup D) }[/math], where [math]\displaystyle{ \sqcup }[/math] is the directed join.[1] When [math]\displaystyle{ 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.[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
A Scott-continuous function is always monotonic, meaning that if [math]\displaystyle{ A \le_{P} B }[/math] for [math]\displaystyle{ A, B \subset P }[/math], then [math]\displaystyle{ 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.[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
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
Footnotes
- ↑ 1.0 1.1 Vickers, Steven (1989). Topology via Logic. Cambridge University Press. ISBN 978-0-521-36062-3.
- ↑ Scott topology in nLab
- ↑ 3.0 3.1 Scott, Dana (1972). "Continuous lattices". in Lawvere, Bill. Toposes, Algebraic Geometry and Logic. Lecture Notes in Mathematics. 274. Springer-Verlag.
- ↑ 4.0 4.1 4.2 4.3 Abramsky, S.; Jung, A. (1994). "Domain theory". in Abramsky, S.; Gabbay, D.M.; Maibaum, T.S.E.. Handbook of Logic in Computer Science. III. Oxford University Press. ISBN 978-0-19-853762-5. http://www.cs.bham.ac.uk/~axj/pub/papers/handy1.pdf.
- ↑ 5.0 5.1 Bauer, Andrej; Taylor, Paul (2009). "The Dedekind Reals in Abstract Stone Duality". Mathematical Structures in Computer Science 19 (4): 757–838. doi:10.1017/S0960129509007695. http://PaulTaylor.EU/ASD/dedras/. 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
References
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