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.
A function between partially ordered sets is Scott-continuous if and only if it is continuous with respect to the Scott topology.
[3] Scott-continuous functions are used in the study of models for lambda calculi[3] and the denotational semantics of computer programs.
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.
[5] For CPO, the cartesian closed category of dcpo's, two particularly notable examples of Scott-continuous functions are curry and apply.