The axiom of choice in metric measure spaces and maximal \(\delta \)-separated sets

Abstract

We show that the Axiom of Countable Choice is necessary and sufficient to prove that the existence of a Borel measure on a pseudometric space such that the measure of open balls is positive and finite implies separability of the space. In this way a negative answer to an open problem formulated in Górka (Am Math Mon 128:84–86, 2020) is given. Moreover, we study existence of maximal \(\delta \)-separated sets in metric and pseudometric spaces from the point of view the Axiom of Choice and its weaker forms.