For example, for any , the set 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.
We verify each of the axioms of an open set:
Clearly itself is Scott open, and is vacuously Scott open.
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.
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.
- 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).