Topometric spaces and perturbations of metric structures

Details Topometric spaces and perturbations of metric structures Topometric spaces and perturbations of metric structures

2008-02-26

arxiv.org/abs/0802.4458v2

Logic and Analysis 1, 3-4 (2008) 235-272

Mathematics

Logic

Scientific paper

10.1007/s11813-008-0009-x

We develop the general theory of \emph{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 \textit{ad hoc} development from \cite{BenYaacov-Usvyatsov:CFO}), as well as of global $\aleph_0$-stability. We conclude with a study of perturbation systems (see \cite{BenYaacov:Perturbations}) 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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