The Scott Topology . Part II

The following propositions are true: (1) Let X be a set and F be a finite family of subsets of X. Then there exists a finite family G of subsets of X such that G ⊆ F and ⋃ G = ⋃ F and for every subset g of X such that g ∈ G holds g 6⊆ ⋃ (G \ {g}). (2) Let S be a 1-sorted structure and X be a subset of the carrier of S. Then −X = the carrier of S if and only if X is empty. (3) Let R be an antisymmetric transitive non empty relational structure with g.l.b.’s and x, y be elements of R. Then ↓(x ⊓ y) = ↓x ∩ ↓y. (4) Let R be an antisymmetric transitive non empty relational structure with l.u.b.’s and x, y be elements of R. Then ↑(x ⊔ y) = ↑x ∩ ↑y.