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 development 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 \({\aleph_0}\)-stability. We conclude with a study of perturbation systems (see Ben Yaacov I, On perturbations of continuous structures, submitted) in the formalism of topometric spaces. In particular, we show how the abstract development applies to \({\aleph_0}\)-stability up to perturbation.
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Ben Yaacov, I. Topometric spaces and perturbations of metric structures. Log Anal 1, 235–272 (2008). https://doi.org/10.1007/s11813-008-0009-x
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DOI: https://doi.org/10.1007/s11813-008-0009-x