Separablilty of metric measure spaces and choice axioms
Document Type
Article
Publication Date
2024
Department/School
Mathematics
Publication Title
Archive for Mathematical Logic
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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} 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 G & oacute;rka (Arch Math Logic 62:735-749, 2023. https://doi.org/10.1007/s00153-023-00868-4).
Link to Published Version
Recommended Citation
Howard, P. (2024). Separablilty of metric measure spaces and choice axioms. Archive for Mathematical Logic, 63(7–8), 987–1003. https://doi.org/10.1007/s00153-024-00931-8
Comments
P. Howard is a faculty member in EMU's Department of Mathematics and Statistics.