# Scott continuity

In mathematics, given two partially ordered sets *P* and *Q*, a function between them is **Scott-continuous** (named after the mathematician Dana Scott) if it preserves all directed suprema, i.e. if 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*: Template:Nowrap beginsup *f*(*D*) = *f*(sup *D*).Template:Nowrap end^{[1]}

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.^{[2]}

Scott-continuous functions show up in the study of models for lambda calculi^{[2]} 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.^{[3]}

A directed complete partial order (dcpo) with the Scott topology is always a Kolmogorov space (i.e., it satisfies the T_{0} separation axiom).^{[3]} However, a dcpo with the Scott topology is a Hausdorff space if and only if the order is trivial.^{[3]} The Scott-open sets form a complete lattice when ordered by inclusion.^{[4]}

For any topological space satisfying the T_{0} separation axiom, 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.^{[3]}

## 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.^{[4]}

For **CPO**, the cartesian closed category of complete partial orders, two particularly notable examples of Scott-continuous functions are curry and apply.^{[5]}

## See also

## Footnotes

- ↑
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*(See theorems 1.2.13, 1.2.14)*