Separablilty of metric measure spaces and choice axioms

Abstract

In set theory without the Axiom of Choice we prove that the assertion “For every metric space (X, d) with a Borel measure \(\mu \) such that the measure of every open ball is positive and finite, (X, d) is separable.’ is implied by the axiom of choice for countable collections of sets and implies the axiom of choice for countable collections of finite sets. We also show that neither implication is reversible in Zermelo–Fraenkel set theory weakend to permit the existence of atoms and that the second implication is not reversible in Zermelo–Fraenkel set theory. This gives an answer to a question of Dybowski and Górka (Arch Math Logic 62:735–749, 2023. https://doi.org/10.1007/s00153-023-00868-4).