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.

中文翻译：

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度量空间和最大 $$\delta $$ 分隔集的选择公理

我们证明可数选择公理是必要且充分的，以证明在伪度量空间上存在 Borel 测度，使得开球的测度为正且有限意味着空间的可分离性。通过这种方式，给出了 Górka（Am Math Mon 128:84–86, 2020）中提出的一个开放问题的否定答案。此外，我们从选择公理及其较弱形式的角度研究了度量和伪度量空间中最大\(\delta \)分离集的存在性。