Definability of groups in ℵ0-stable metric structures

Abstract

AbstractWe prove that in a continuous ℵ0-stable theory every type-definable group is definable. The two main ingredients in the proof are:(i) Results concerning Morley ranks (i.e., Cantor-Bendixson ranks) from [Ben08], allowing us to prove the theorem in case the metric is invariant under the group action; and(ii) Results concerning the existence of translation-invariant definable metrics on type-definable groups and the extension of partial definable metrics to total ones.