Equivalence Relations Invariant under Group Actions

We study, in an abstract context, equivalence relations which are invariant under group actions. More precisely, we fix a transformation group, and we study the orbital equivalence relations (i.e. orbit equivalence relations of normal subgroups) and a wider class of weakly orbital equivalence relations. For these sorts of relations we show (under some additional assumptions) that if each class is well-behaved, then so is the class space and the relation as a whole (where well-behaved means type-definable or closed for the classes and the relation and Hausdorff for the class space). We apply these conclusions in model theory to generalise a recent result tying type-definability and smoothness of invariant equivalence relations. We also obtain analogous results for typedefinable actions (in model theory), as well as continuous actions of compact groups, or, more generally, proper group actions.