Local stability in infinitary real-valued logic

In this paper we study the stability of formulas in the infinitary extension of Keisler’s model theory of real-valued structures. We show that four classical characterizations of stability carry over to this logic, with the types realized in elementary extensions in place of all types: the absence of the order property, the commutation of iterated limits of the values of the formula along sequences of elements, the existence of a function on pairs of types that extends the values of the formula on pairs of elements and is continuous in each variable, and the definability of types. The characterizations follow from a double limit criterion for prescribed limits, which we prove directly in the spirit of Arhangel’skii’s Cₚ-theory; the Čech completeness of the spaces of Henkin types, which yields a model existence theorem in place of compactness, describes the realized types, for relational vocabularies, as analytic sets.