Topometric spaces and perturbations of metric structures

TL;DR

This paper introduces the theory of topometric spaces, combining topology and metric structures, and applies it to analyze stability and perturbations in continuous logic, providing a unified framework for local and global stability analysis.

Contribution

It develops a comprehensive theory of topometric spaces and extends Cantor-Bendixson analysis to study local and global stability in continuous logic.

Findings

Topometric spaces unify topology and metric structures.

A Cantor-Bendixson analysis for topometric spaces is established.

The framework applies to stability and perturbation analysis in continuous logic.

Abstract

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.