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.
"Definability of groups in ℵ₀-stable metric structures." J. Symbolic Logic 75 (3) 817 - 840, September 2010. https://doi.org/10.2178/jsl/1278682202