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.
U an upper set: ↑U=U, and
-
2.
if D is a directed set
with ⋁D∈U, then there is a y∈D such that (↑y)∩D⊆U.
Condition 2 is equivalent to saying that U has non-empty intersection
with D whenever D is directed and its supremum
is in U.
For example, for any x∈P, the set U(x):= is Scott open: if , then there is with . Since , . So , or that is upper. If is directed and for all , then as well. Therefore, implies for some . Hence is Scott open.
The collection of all Scott open sets of is a topology
, called the Scott topology of , named after its inventor Dana Scott. Let us prove that is indeed a topology:
Proof.
We verify each of the axioms of an open set:
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•
Clearly itself is Scott open, and is vacuously Scott open.
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•
Suppose and are Scott open. Let and . Then for some , . Since , and . This means , so is an upper set. Next, if is directed with , then, . So there are with and . Since is directed, there is such that . So . This means that is Scott open.
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•
Suppose are open and an index set
. Let and . So for some . Since for some , as is upper. Hence , or that is upper. Next, suppose is directed with . Then for some . Since is Scott open, there is with , so is Scott open.
Since the Scott open sets satisfy the axioms of a topology, is a topology on . ∎
Examples. If is the unit interval: , then is a complete chain, hence a dcpo. Any Scott open set has the form if , or . If , the unit square, then is a dcpo as it is already a continuous lattice. The Scott open sets of are any upper subset of that is also an open set in the usual sense.
References
-
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 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 |