Topometric spaces and perturbations of metric structures. (English) Zbl 1180.03040
Summary: 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 [I. Ben Yaacov and A. Usvyatsov, “Continuous first-order logic and local stability”, Trans. Am. Math. Soc. (to appear)], as well as of global \(\aleph_0\)-stability. We conclude with a study of perturbation systems (see [I. Ben Yaacov, “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.
MSC:
| 03C95 | Abstract model theory |
| 03C45 | Classification theory, stability, and related concepts in model theory |
| 03C90 | Nonclassical models (Boolean-valued, sheaf, etc.) |
| 54H99 | Connections of general topology with other structures, applications |
References:
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