Extreme types and extremal models

Abstract

In the affine fragment of continuous logic, type spaces are compact convex sets. I study some model theoretic properties of extreme types. It is proved that every complete theory T has an extremal model, i.e. a model which realizes only extreme types. Extremal models form an elementary class in the full continuous logic sense if and only if the set of extreme n-types is closed in $S_n\left(T\right)$ for each n. Also, some applications are given in the special cases where the theory has a compact or first order model.