  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.
