Definability of groups in ℵ₀-stable metric structures

Abstract

We prove that in a continuous ℵ₀-stable theory every type-definable group is definable. The two main ingredients in the proof are:

1. 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

2. Results concerning the existence of translation-invariant definable metrics on type-definable groups and the extension of partial definable metrics to total ones.