On BPI Restricted to Boolean Algebras of Size Continuum

Eric Hall; Kyriakos Keremedis

Bulletin of the Polish Academy of Sciences. Mathematics (2013)

  • Volume: 61, Issue: 1, page 9-21
  • ISSN: 0239-7269

Abstract

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(i) The statement P(ω) = “every partition of ℝ has size ≤ |ℝ|” is equivalent to the proposition R(ω) = “for every subspace Y of the Tychonoff product 2 ( ω ) the restriction |Y = Y ∩ B: B ∈ of the standard clopen base of 2 ( ω ) to Y has size ≤ |(ω)|”. (ii) In ZF, P(ω) does not imply “every partition of (ω) has a choice set”. (iii) Under P(ω) the following two statements are equivalent: (a) For every Boolean algebra of size ≤ |ℝ| every filter can be extended to an ultrafilter. (b) Every Boolean algebra of size ≤ |ℝ| has an ultrafilter.

How to cite

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Eric Hall, and Kyriakos Keremedis. "On BPI Restricted to Boolean Algebras of Size Continuum." Bulletin of the Polish Academy of Sciences. Mathematics 61.1 (2013): 9-21. <http://eudml.org/doc/281328>.

@article{EricHall2013,
abstract = {(i) The statement P(ω) = “every partition of ℝ has size ≤ |ℝ|” is equivalent to the proposition R(ω) = “for every subspace Y of the Tychonoff product $2^\{(ω)\}$ the restriction |Y = Y ∩ B: B ∈ of the standard clopen base of $2^\{(ω)\}$ to Y has size ≤ |(ω)|”. (ii) In ZF, P(ω) does not imply “every partition of (ω) has a choice set”. (iii) Under P(ω) the following two statements are equivalent: (a) For every Boolean algebra of size ≤ |ℝ| every filter can be extended to an ultrafilter. (b) Every Boolean algebra of size ≤ |ℝ| has an ultrafilter.},
author = {Eric Hall, Kyriakos Keremedis},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {weak axioms of choice; filters; ultrafilters; Boolean prime ideal theorem; regular open sets},
language = {eng},
number = {1},
pages = {9-21},
title = {On BPI Restricted to Boolean Algebras of Size Continuum},
url = {http://eudml.org/doc/281328},
volume = {61},
year = {2013},
}

TY - JOUR
AU - Eric Hall
AU - Kyriakos Keremedis
TI - On BPI Restricted to Boolean Algebras of Size Continuum
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2013
VL - 61
IS - 1
SP - 9
EP - 21
AB - (i) The statement P(ω) = “every partition of ℝ has size ≤ |ℝ|” is equivalent to the proposition R(ω) = “for every subspace Y of the Tychonoff product $2^{(ω)}$ the restriction |Y = Y ∩ B: B ∈ of the standard clopen base of $2^{(ω)}$ to Y has size ≤ |(ω)|”. (ii) In ZF, P(ω) does not imply “every partition of (ω) has a choice set”. (iii) Under P(ω) the following two statements are equivalent: (a) For every Boolean algebra of size ≤ |ℝ| every filter can be extended to an ultrafilter. (b) Every Boolean algebra of size ≤ |ℝ| has an ultrafilter.
LA - eng
KW - weak axioms of choice; filters; ultrafilters; Boolean prime ideal theorem; regular open sets
UR - http://eudml.org/doc/281328
ER -

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