Topometric spaces and perturbations of metric structures

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.

References in corpus (3)

• On perturbations of continuous structures
• Modular functionals and perturbations of Nakano spaces
• On perturbations of Hilbert spaces and probability algebras with a generic automorphism

Cited by in corpus (14)

• On theories of random variables
• On perturbations of continuous structures
• Modular functionals and perturbations of Nakano spaces
• Definability of groups in ${\aleph }_0$-stable metric structures
• Duality for optimal couplings in free probability
• Enforceable operator algebras
• Topological dynamics beyond Polish groups
• Lipschitz functions on topometric spaces
• On perturbations of Hilbert spaces and probability algebras with a generic automorphism
• Complexity of distances: Theory of generalized analytic equivalence relations
• ${\aleph }_0$-categorical Banach spaces contain ${\ell }_p$ or $c_0$
• Continuous stable regularity
• On dense orbits in the space of subequivalence relations
• Topometric characterization of type spaces in continuous logic