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