# Scott continuity

In mathematics, given two partially ordered sets P and Q, a function f: PQ 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, 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 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: xy 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]

4. ^ 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. `|volume=` has extra text (help)