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. ${\displaystyle \sqcup f[D]=f(\sqcup D)}$, where ${\displaystyle \sqcup }$ is the directed join.[1] When ${\displaystyle Q}$ 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 ${\displaystyle A\leq _{P}B}$ for ${\displaystyle A,B\subset P}$, then ${\displaystyle f(A)\leq _{Q}f(B)}$.

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

• Alexandrov topology
• Upper topology

Footnotes

1. Vickers, Steven (1989). Topology via Logic. Cambridge University Press. ISBN 978-0-521-36062-3.
2. Scott topology at the nLab
3. Scott, Dana (1972). "Continuous lattices". In Lawvere, Bill (ed.). Toposes, Algebraic Geometry and Logic. Lecture Notes in Mathematics. Vol. 274. Springer-Verlag.
4. 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.
5. 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.
6. Barendregt, H.P. (1984). The Lambda Calculus. North-Holland. ISBN 978-0-444-87508-2. (See theorems 1.2.13, 1.2.14)
7. 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

• "Scott Topology". PlanetMath.