Topometric spaces and perturbations of metric structures

We develop the general theory of topometric spaces, i.e., topological spaces equipped with a well-behaved lower semi-continuous metric function. Spaces of global and local types in continuous logic are the motivating examples for the study of such spaces. In particular, we develop a theory of Cantor-Bendixson analysis of topometric spaces, which can serve as a basis for the study of local stability (extending the ad hoc development from [BU]), as well as of global א0-stability. We conclude with a study of perturbation systems (see [Benb]) in the formalism of topometric spaces. In particular, we show how the abstract development applies to א0-stability up to perturbation.