Topometric spaces and perturbations of metric structures |
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Authors: | Itaï Ben Yaacov |
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Affiliation: | 1. Université de Lyon, Université Lyon 1, Institut Camille Jordan, CNRS UMR 5208, 43 boulevard du 11 novembre 1918, 69622, Villeurbanne Cedex, France
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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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