Stability theory for topological logic, with applications to topological modules

Abstract

In this paper I show how to develop stability theory within the context of the topological logic first introduced by McKee [Mc 76], Garavaglia [G 78] and Ziegler [Z 76]. I then study some specific applications to topological modules; in particular I prove two quantifier élimination theorems, one a generalization of a result of Garavaglia.In the first section I present a summary of basic results on topological model theory, mostly taken from the book of Flum and Ziegler [FZ 80]. This is done primarily to fix notation, but I also introduce the notion of anLt-elementary substructure. The important point with this concept, as with many others, appears to be to allow only individuals to appear as parameters, not open sets.In the second section I begin the study of stability theory forLt. I first develop a translation of the topological languageLtinto an ordinary first-order languageL*. The first main theorem is (2.3), which shows that the translation is faithful to the model-theoretic content ofLt, and provides the necessary tools for studyingLttheories in the context of ordinary first-order logic. The translation allows me to considerindividual stability theoryforLt: the stability-theoretic study of those types ofLtin which only individual variables occur freely and in which only individuals occur as parameters. I originally developed this stability theory entirely withinLt; the fact that the theorems and their proofs were virtually identical to those in ordinary first order logic suggested the reduction fromLttoL*.