 
Summary: Chapter 3
Continuous and Algebraic
Domains
(Notes for CS 819  April 17, 2000  c
fl2000 John C. Reynolds)
3.1 Topology
Let P be any dcpo. Then U ` P is said to be open iff
1. U is upward closed, i.e., if x 2 U and x v y, then y 2 U ;
2. For all directed D ` P , if F
D 2 U , then there is an x 2 D such that
x 2 U .
On the other hand, C ` P is said to be closed iff
1. C is downward closed, i.e., if y 2 C and x v y, then x 2 C;
2. For all directed D ` P , if D ` C, then F
D 2 C.
(Note that the second conditions in both of these definitions are trivially true
for uninteresting directed sets.) It is easily seen that S ` P is open (closed)
iff P \Gamma S is closed (open), that unions and finite intersections of open sets are
open, and that both P and the empty set are both open and closed. If P has
a least element or a greatest element, then these are the only subsets that
