# 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. , where is the directed join.[1] When is the poset of truth values, i.e. Sierpiński space, then Scott-continuous functions are characteristic functions, and thus Sierpiński space is the classifying topos 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 show up in the study of models for lambda calculi[3] and the denotational semantics of computer programs.

## Properties

A Scott-continuous function is always monotonic.

A subset of a partially ordered set 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 T_{0} 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

- Vickers, Steven (1989).
*Topology via Logic*. Cambridge University Press. ISBN 978-0-521-36062-3. - Scott topology in
*nLab* - Scott, Dana (1972). "Continuous lattices". In Lawvere, Bill (ed.).
*Toposes, Algebraic Geometry and Logic*. Lecture Notes in Mathematics.**274**. Springer-Verlag. - 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. - 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. 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