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The independence of Stone's Theorem from the Boolean Prime Ideal Theorem


Author: Samuel M. Corson
Journal: Proc. Amer. Math. Soc. 148 (2020), 5381-5386
MSC (2010): Primary 03E25, 54A35, 54E35, 54D20
DOI: https://doi.org/10.1090/proc/15164
Published electronically: September 18, 2020
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Abstract: We give a permutation model in which Stone's theorem (every metric space is paracompact) is false and the Boolean Prime Ideal Theorem (every ideal in a Boolean algebra extends to a prime ideal) is true. The erring metric space in our model attains only rational distances and is not metacompact. Transfer theorems give the comparable independence in the Zermelo-Fraenkel setting, answering a question of Good, Tree, and Watson.


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Additional Information

Samuel M. Corson
Affiliation: Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, 28049 Madrid, Spain
MR Author ID: 1133429
ORCID: 0000-0003-0050-2724
Email: sammyc973@gmail.com

DOI: https://doi.org/10.1090/proc/15164
Keywords: Paracompact, metacompact, metric space, Boolean Prime Ideal Theorem
Received by editor(s): February 17, 2020
Received by editor(s) in revised form: April 9, 2020, April 14, 2020, and April 15, 2020
Published electronically: September 18, 2020
Additional Notes: This work was supported by ERC grant PCG-336983 and by the Severo Ochoa Programme for Centres of Excellence in R&D SEV-20150554.
Communicated by: Heike Mildenberger
Article copyright: © Copyright 2020 American Mathematical Society