Library.
3 verifications about Axiom of Choice Axiom of Choice ×
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True 10/10
“The Boolean prime ideal theorem is equivalent, in Zermelo–Fraenkel set theory without the axiom of choice, to Tychonoff's theorem for compact Hausdorff spaces.”
The stated equivalence is a standard result of choice theory. In ZF, the Boolean prime ideal theorem—equivalently, the ultrafilter lemma—proves that products of compact Hausdorff spaces are compact, and that restricted form of Tychonoff’s theorem implies the Boolean prime ideal theorem. Authoritative mathematical sources support both directions.
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True 9/10
“In set theory without the axiom of choice but assuming the Boolean prime ideal theorem, every product of nonempty compact Hausdorff spaces is nonempty.”
The Boolean prime ideal theorem suffices to guarantee that every product of nonempty compact Hausdorff spaces has a point. This is a standard choice-principle equivalence in ZF and is supported by direct and independent mathematical sources. It does not require the full axiom of choice.
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Mostly False 4/10
“Every infinite set contains a countably infinite subset.”
As stated, the theorem is too broad. In ordinary ZFC-based mathematics, every infinite set does contain a countably infinite subset, but without some form of the Axiom of Choice this is not generally true. In ZF, there can be infinite Dedekind-finite sets with no countably infinite subset, so the omitted assumption materially changes the claim.