Definability of groups in ${\aleph }_0$-stable metric structures

Abstract

We prove that in a continuous ${\aleph }_0$-stable theory every type-definable group is definable. The two main ingredients in the proof are: \begin{enumerate} \item Results concerning Morley ranks (i.e., Cantor-Bendixson ranks) from \cite{BenYaacov:TopometricSpacesAndPerturbations}, allowing us to prove the theorem in case the metric is invariant under the group action; and \item Results concerning the existence of translation-invariant definable metrics on type-definable groups and the extension of partial definable metrics to total ones. \end{enumerate}

References in corpus (1)

• Topometric spaces and perturbations of metric structures

Cited by in corpus (3)

• On theories of random variables
• Locally compact groups which are separably categorical structures
• Star sorts, Lelek fans, and the reconstruction of non-$\aleph \_0$-categorical theories in continuous logic