Scott topology

Let P be a dcpo. A subset U of P is said to be Scott open if it satisfies the following two conditions:

  1. 1.

    U an upper set: U=U, and

  2. 2.

    if D is a directed setMathworldPlanetmath with DU, then there is a yD such that (y)DU.

Condition 2 is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to saying that U has non-empty intersectionMathworldPlanetmath with D whenever D is directed and its supremumMathworldPlanetmathPlanetmath is in U.

For example, for any xP, the set U(x):=P-(x) is Scott open: if yU(x), then there is zU(x) with zy. Since zx, yx. So yU(x), or that U(x) is upper. If D is directed and ex for all eD, then d:=Dx as well. Therefore, dU(x) implies eU(x) for some eD. Hence U(x) is Scott open.

The collectionMathworldPlanetmath σ(P) of all Scott open sets of P is a topologyMathworldPlanetmathPlanetmath, called the Scott topology of P, named after its inventor Dana Scott. Let us prove that σ(P) is indeed a topology:


We verify each of the axioms of an open set:

  • Clearly P itself is Scott open, and is vacuously Scott open.

  • Suppose U and V are Scott open. Let W=UV and bW. Then for some aW, ab. Since aUV, bU=U and bV=V. This means bW, so W is an upper set. Next, if D is directed with DW, then, DUV. So there are y,zD with (y)DU and (z)DV. Since D is directed, there is tD such that t(y)(z). So (t)D(y)(z)D=((y)D)((z)D)UV=W. This means that W is Scott open.

  • Suppose Ui are open and iI an index setMathworldPlanetmathPlanetmath. Let U={UiiI} and bU. So ab for some aU. Since aUi for some iI, bUi=Ui as Ui is upper. Hence bUiU, or that U is upper. Next, suppose D is directed with DU. Then DUi for some iI. Since Ui is Scott open, there is yD with (y)DUiU, so U is Scott open.

Since the Scott open sets satisfy the axioms of a topology, σ(P) is a topology on P. ∎

Examples. If P is the unit interval: P=[0,1], then P is a complete chain, hence a dcpo. Any Scott open set has the form (a,1] if 0<a1, or [0,1]. If P=[0,1]×[0,1], the unit square, then P is a dcpo as it is already a continuous lattice. The Scott open sets of P are any upper subset of P that is also an open set in the usual sense.


  • 1 G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, D. S. Scott, ContinuousPlanetmathPlanetmath Lattices and Domains, Cambridge University Press, Cambridge (2003).
Title Scott topology
Canonical name ScottTopology
Date of creation 2013-03-22 16:49:29
Last modified on 2013-03-22 16:49:29
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Definition
Classification msc 06B35
Defines Scott open