The Scott Topology . Part II 1

Mizar formalization of pp. 105–108 of [10] which continues [27]. We found a simplification for the proof of Corollary 1.15, in the last case, see the proof in the Mizar article for details. One can prove the following propositions: (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 ⊆ (G \ {g}). (2) For every 1-sorted structure S and for every subset X of S holds X c = the carrier of S iff 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. (5) Let L be a complete antisymmetric non empty relational structure and X be a lower subset of L. If sup X ∈ X, then X = ↓sup X. (6) Let L be a complete antisymmetric non empty relational structure and X be an upper subset of L. If inf X ∈ X, then X = ↑inf X. (7) Let R be a non empty reflexive transitive relational structure and x, y be elements of R. Then x y if and only if ↑y ⊆ ↑ ↑ x. (8) Let R be a non empty reflexive transitive relational structure and x, y be elements of R. Then x y if and only if ↓x ⊆ ↓ ↓ y.