In mathematics, a function is Scott-continuous if it is continuous with respect to the Scott topology. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Partial plot of a function f. ... Dana Stewart Scott (born 1932) is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, California. ... This is a glossary of some terms used in various branches of mathematics that are related to the fields of order, lattice, and domain theory. ...

In the case where the domain and range are both partially ordered sets equipt with the Scott topology this is equivalent to specifying that the function is monotonic and preserves all directed suprema. In mathematics, the domain of a function is the set of all input values to the function. ... In mathematics, the range of a function is the set of all output values produced by that function. ... In mathematics, especially order theory, a partially ordered set (or poset for short) is a set equipped with a partial order relation. ... In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ... In the mathematical area of order theory, one often speaks about functions that preserve certain limits, i. ... In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ...

A monotone 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 In mathematics, a directed set is a set A together with a binary relation ≤ having the following properties: a ≤ a for all a in A (reflexivity) if a ≤ b and b ≤ c, then a ≤ c (transitivity) for any two a and b in A, there...

sup D in P,

the set

{fx | x in D}

has the supremum

f(sup D) in Q.

See also: Glossary of order theory This is a glossary of some terms used in various branches of mathematics that are related to the fields of order, lattice, and domain theory. ...