Monadic Characterizations in Nonstandard Topology

Abstract

We intend to present a syntactical means of producing the common nonstandard (or monadic) characterizations of the basic notions of topology. I n addition to particular examples (e.g., of Hausdorff and regularity), we shall indicate a connection with ROBINSON'S 9-topology and with the so-called "Cauchy Principle."Let L~Y be a topological basis on a set A ; i.e.,The language 9 has relations = (between points but not between sets) and E (between a point and a set). Also, 9 has the connectives 1 , A, v and quantifiers of the forms (Vz E A ) and (32 E A ) over points and (VX E 8%) and (3X E gz) over basic neighborhoods of points. The *-transform *pl of a formula v of 2 ' is produced by starring each constant (and A , LBJ, av, . . .) which occurs in pl.Lemma.(1) If X occurs only negatively in pl(X) and if G, H E *LiYp such that H s clr, then *y[G] + * y [ H ] .(2) If X occurs only negatively in y ( X ) and y ( X ) , then [(3x E g p ) *y(*x) A (3x E gp) *W(*X)] 4 (3x E gp) [*y(*X) A *W(*x)]. Proof. Part (1) follows by a straightforward induction on the complexity of y . A *y(*X)]. (This lemma occurs in a different setting in [8].)Theorem. If X occurs only negatively in a formula y ( X ) of 9, then (3X E g p ) v ( X ) c, *v[p(p)]. If X occurs only positively in a formula y ( X ) of 2, then (VX E Bp) y ( X ) c, *p[p(p)].Proof. The second version will follow from the first by contraposition. To prove the first, suppose y ( X ) is Q y ( X , Y,, . . .) where Q is a string of quantifiers, Y l , . . .are the set variables quantified in Q, and ly is quantifier-free and in disjunctive normal form. The theorem will follow from the equivalence of the following three sentences: