Some consequences of Rado’s selection lemma

Abstract

We prove in set theory without the Axiom of Choice, that Rado's selection lemma ( $${\mathbf{RL}}$$ ) implies the Hahn-Banach axiom. We also prove that $${\mathbf{RL}}$$ is equivalent to several consequences of the Tychonov theorem for compact Hausdorff spaces: in particular, $${\mathbf{RL}}$$ implies that every filter on a well orderable set is included in a ultrafilter. In set theory with atoms, the "Multiple Choice" axiom implies $${\mathbf{RL}}$$ .