Galois groups as quotients of Polish groups

Abstract

We present the (Lascar) Galois group of any countable theory as a quotient of a compact Polish group by an [Formula: see text] normal subgroup: in general, as a topological group, and under NIP, also in terms of Borel cardinality. This allows us to obtain similar results for arbitrary strong types defined on a single complete type over [Formula: see text]. As an easy conclusion of our main theorem, we get the main result of [K. Krupiński, A. Pillay and T. Rzepecki, Topological dynamics and the complexity of strong types, Israel J. Math. 228 (2018) 863–932] which says that for any strong type defined on a single complete type over [Formula: see text], smoothness is equivalent to type-definability. We also explain how similar results are obtained in the case of bounded quotients of type-definable groups. This gives us a generalization of a former result from the paper mentioned above about bounded quotients of type-definable subgroups of definable groups.