Permutation models and topological groups
Fraenkel-Mostowski models are the classical device for proving independence results for the axiom of choice (AC) in set theory with atoms. In these models the validity of various forms of AC is characterized in terms of the topological automorphism group. Thus a permutation model satisfies Levy's multiple choice axiom (MC), if and only if its group of \(\in\)-automorphisms is locally compact. This implies that in all permutation models MC together with the countable axiom of choice is equivalent to AC. Particular emphasis rests on the investigation of models with a proper class of atoms.
scientific article; zbMATH DE number 4008390