Topometric spaces and perturbations of metric structures

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 $\mathit{\text{ad}}\mathit{\text{hoc}}$ development from \cite{BenYaacov-Usvyatsov:CFO}), as well as of global ${\mathrm{\aleph }}_0$-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 ${\mathrm{\aleph }}_0$-stability up to perturbation.