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BPI guarantees a point in every product of nonempty compact Hausdorff spaces.

The Claim

In set theory without the axiom of choice but assuming the Boolean prime ideal theorem, every product of nonempty compact Hausdorff spaces is nonempty.

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The Short Version

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.

Caveats

  • Compactness alone does not imply nonemptiness because the empty space is compact.
  • The result is restricted to compact Hausdorff factors, not arbitrary nonempty spaces or sets.
  • Counterexamples arising in bare ZF are irrelevant unless their models also satisfy BPI.

The Receipts

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

    doi.org

  2. Powers of 2

    doi.org

  3. Commentationes Mathematicae Universitatis Carolinae

    dml.cz

  4. The Axiom of Choice - Stanford Encyclopedia of Philosophy

    plato.stanford.edu

  5. Axiom of Choice - Stanford Encyclopedia of Philosophy

    plato.stanford.edu

  6. Tychonoff theorem in nLab

    ncatlab.org

  7. Boolean prime ideal theorem in nLab

    ncatlab.org

  8. Several results on compact metrizable spaces in $$\mathbf {ZF}$$

    doi.org

  9. Prime ideal theorem - Encyclopedia of Mathematics

    encyclopediaofmath.org

  10. ultrafilter theorem in nLab

    ncatlab.org

+ 21 more sources — see the full list on Lenz

Filed Under

Axiom Of ChoiceBoolean Prime Ideal TheoremCompact Hausdorff Spaces

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