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In ZF, compact Hausdorff Tychonoff equals the Boolean prime ideal theorem.

The Claim

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.

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The Short Version

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.

Caveats

  • The Hausdorff restriction is essential; Tychonoff’s theorem for all compact spaces has a stronger relationship to the full axiom of choice.
  • The equivalence assumes the standard open-cover definition of compactness and the usual product topology in ZF.
  • Several listed sources are duplicates or informal references, but the result is independently supported by authoritative mathematical literature.

The Receipts

  1. Commentationes Mathematicae Universitatis Carolinae

    dml.cz

  2. The Axiom of Choice (Stanford Encyclopedia of Philosophy)

    plato.stanford.edu

  3. AN ALTERNATIVE PROOF OF THE TYCHONOFF THEOREM

    doi.org

  4. The Axiom of Choice - Stanford Encyclopedia of Philosophy

    plato.stanford.edu

  5. Compactness and the axiom of choice | Applied Categorical Structures

    link.springer.com

  6. The Axiom of Choice - Stanford Encyclopedia of Philosophy

    plato.stanford.edu

  7. Kelley's specialization of Tychonoff's Theorem is equivalent to the Boolean Prime Ideal TheoremAll

    impan.pl

  8. Kelley's specialization of Tychonoff's Theorem is equivalent to the Boolean Prime Ideal Theorem

    geodesic.mathdoc.fr

  9. Kelley's specialization of Tychonoff's Theorem is equivalent to ... - EuDML

    eudml.org

  10. prime ideal theorem in nLab

    ncatlab.org

+ 16 more sources — see the full list on Lenz

Filed Under

Axiom Of ChoiceBoolean Prime Ideal TheoremTychonoff's Theorem for Compact Hausdorff SpacesZermelo–Fraenkel Set Theory

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