Equivalence relations invariant under group actions

TL;DR

This paper broadens the understanding of invariant equivalence relations under group actions, showing type-definability and smoothness are equivalent in a wider context, with applications in both model theory and topology.

Contribution

It extends previous results by establishing the equivalence of type-definability and smoothness for invariant relations in a more general setting, including topological cases.

Findings

Type-definability and smoothness are equivalent in a broader class of invariant relations.

Results apply to both model-theoretic and topological contexts.

The work generalizes prior specific cases to a wider framework.

Abstract

We extend some recent results about bounded invariant equivalence relations and invariant subgroups of definable groups: we show that type-definability and smoothness are equivalent conditions in a wider class of relations than heretofore considered, which includes all the cases for which the equivalence was proved before. As a by-product, we show some analogous results in purely topological context (without direct use of model theory).