Scott continuity

A monotone function f : P → Q between posets P and Q is Scott-continuous if, for every directed set D that has a supremum

sup D in P,

the set

{fx | x in D}

has the supremum

f(sup D) in Q.

Stated differently, a Scott-continuous function is one that preserves all directed suprema. This is in fact equivalent to being continuous with respect to the Scott topology on the respective posets.

See also: : Glossary of order theory