# 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 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 show up 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 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[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*n*Lab- ^
^{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