Scott continuous

Let P1,P2 be two dcpos ( A function f:P1P2 is said to be Scott continuous if for any directed setMathworldPlanetmath DP1, f(D)=f(D).

First, observe that f is monotoneMathworldPlanetmathPlanetmath. If ab, then f(b)=f({a,b})={f(a),f(b)}, so that f(a)f(b). As a result, if D is directed, so is f(D).

Proposition 1.

f:P1P2 is Scott continuous iff it is continuousMathworldPlanetmathPlanetmath when P1 and P2 are equipped with the Scott topologies.

Before proving this, let’s make one additional observation:

Lemma 1.

If f is continuous (under Scott topologies), then f is monotone.


Suppose abP1. We wish to show that f(a)f(b), or f(a)f(b). Assume the contrary. Consider U=P2-f(b). Then f(a)U and U is Scott open, hence af-1(U) is Scott open also. Since ab and f-1(U) is upper, bf-1(U), which implies f(b)U=P2-f(b), a contradictionMathworldPlanetmathPlanetmath. Therefore, f(a)f(b). ∎

Now the proof of the propositionPlanetmathPlanetmathPlanetmath.


Suppose first that f is Scott continuous. Take an open set UP2. We want to show that V:=f-1(U) is open in P1. In other words, V is upper and that V has non-empty intersectionMathworldPlanetmath with any directed set DP1 whenever its supremumMathworldPlanetmathPlanetmath D lies in V. If aV, then some bV with ba, which implies f(b)f(a). Since f(b)U, f(a)U=U, so af-1(U)=V, V is upper. Now, suppose DV. So f(D)=f(D)U. Since f(D) is directed, there is yf(D)U, which means there is xP1 such that f(x)=y and xDV. This shows that V is Scott open.

Conversely, suppose f is continuous (inversePlanetmathPlanetmathPlanetmathPlanetmath of a Scott open set is Scott open). Let D be a directed subset of P1 and let d=D. We want to show that f(d)=f(D). First, for any eD, we have that ed so that f(e)f(d) since f is monotone. This shows f(D)f(d). Now suppose r is any upper boundMathworldPlanetmath of f(D). We want to show that f(d)r, or f(d)r. Assume not. Then f(d) lies in U:=P2-r, a Scott open set. So D=df-1(U), also Scott open, which implies some eD with ef-1(U), or f(e)U. This means f(e)r, a contradiction. Thus f(d)r, and the proof is completePlanetmathPlanetmathPlanetmathPlanetmath. ∎

Remark. This notion of continuity is attributed to Dana Scott when he was trying to come up with a model for the formal system of untyped lambda calculusMathworldPlanetmath.


  • 1 G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, D. S. Scott, Continuous Lattices and Domains, Cambridge University Press, Cambridge (2003).
Title Scott continuous
Canonical name ScottContinuous
Date of creation 2013-03-22 16:49:53
Last modified on 2013-03-22 16:49:53
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 6
Author CWoo (3771)
Entry type Definition
Classification msc 06B35