Maximal \delta-separated sets in separable metric spaces and weak forms of choice
Micha{\l} Dybowski, Przemyslaw G\'orka, Paul Howard
TL;DR
The paper investigates the logical strength of the statement that every separable pseudometric space has a maximal -separated set, showing it implies the axiom of choice for countable families, and explores related results.
Contribution
It demonstrates that a natural statement about maximal -separated sets in separable pseudometric spaces implies the axiom of choice for countable families, linking topology and set theory.
Findings
The statement implies the axiom of choice for countable families.
Several related results connecting maximal -separated sets and choice principles.
Answers a question posed by Dybowski and Grka in their 2023 paper.
Abstract
We show that the statement ``In every separable pseudometric space there is a maximal non-strictly \delta-separated set.'' implies the axiom of choice for countable families of sets. This gives answers to a question of Dybowski and G\'{o}rka in [M. Dybowski and P. G\'{o}rka, The axiom of choice in metric measure spaces and maximal \delta-separated sets, Archive for Mathematical Logic 62, 735-749, 2023.]. We also prove several related results.
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