structures

Abstract We develop the general theory of topometric spaces, i.e., topological spaces equipped with a well-behaved lower semi-continuous metric. Spaces of global and local types in continuous logic are the motivating examples for the study of such spaces. In particular, we develop Cantor-Bendixson analysis of topometric spaces, which can serve as a basis for the study of local stability (extending the ad hoc devel-opment in Ben Yaacov I and Usvyatsov A, Continuous first order logic and local stability. Trans Am Math Soc, in press), as well as of global ℵ0-stability. We conclude with a study of perturbation systems (see Ben Yaacov I, On perturbations of contin-uous structures, submitted) in the formalism of topometric spaces. In particular, we show how the abstract development applies to ℵ0-stability up to perturbation