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

Corson, Samuel

doi:10.1090/proc/15164

The independence of Stone’s Theorem from the Boolean Prime Ideal Theorem
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by Samuel M. Corson

Proc. Amer. Math. Soc. 148 (2020), 5381-5386

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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