The Tychonoff product theorem for compact Hausdorff spaces does not imply the axiom of choice: A new Proof. equivalent propositions.
B. R. Salinas, F. Bombal (1973)
Collectanea Mathematica
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B. R. Salinas, F. Bombal (1973)
Collectanea Mathematica
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D. Edwards (1975)
Fundamenta Mathematicae
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J. Halperin (1964)
Fundamenta Mathematicae
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Miroslav Repický (2015)
Commentationes Mathematicae Universitatis Carolinae
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We present a proof of the Boolean Prime Ideal Theorem in a transitive model of ZF in which the Axiom of Choice does not hold. We omit the argument based on the full Halpern-Läuchli partition theorem and instead we reduce the proof to its elementary case.
Eric Schechter (2006)
Fundamenta Mathematicae
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The principle that "any product of cofinite topologies is compact" is equivalent (without appealing to the Axiom of Choice) to the Boolean Prime Ideal Theorem.
G. Gardiner (1974)
Fundamenta Mathematicae
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J. Łoś, Czesław Ryll-Nardzewski (1955)
Fundamenta Mathematicae
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J. Bell (1988)
Fundamenta Mathematicae
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J. Łoś (1957)
Fundamenta Mathematicae
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Marcel Erné (1997)
Commentationes Mathematicae Universitatis Carolinae
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We study several choice principles for systems of finite character and prove their equivalence to the Prime Ideal Theorem in ZF set theory without Axiom of Choice, among them the Intersection Lemma (stating that if is a system of finite character then so is the system of all collections of finite subsets of meeting a common member of ), the Finite Cutset Lemma (a finitary version of the Teichm“uller-Tukey Lemma), and various compactness theorems. Several implications between these...