Scott continuity

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In mathematics, a function is Scott-continuous if it is continuous with respect to the Scott topology.

In the case where the domain and range are both partially ordered sets equipped with the Scott topology this is equivalent to specifying that the function preserves all directed suprema. This also implies that the function is monotonic.

A function

f : P → Q

between partially ordered sets P and Q preserves all directed suprema if for every directed set D that has a supremum

sup D in P,

the set

{f(x) | x in D}

has the supremum

f(sup D) in Q.

See also: Glossary of order theory