Triple

Target entity: Ultrafilter lemma
Context triple: [Tychonoff theorem for products of compact spaces, relatedTo, Ultrafilter lemma]

• A. Hausdorff maximal principle
  The Hausdorff maximal principle is a foundational result in set theory and order theory stating that every partially ordered set contains a maximal totally ordered subset (a maximal chain), and it is equivalent to the axiom of choice.
• B. Kuratowski–Zorn lemma (attribution as Zorn’s lemma variant)
  The Kuratowski–Zorn lemma is a fundamental result in set theory and order theory, equivalent to the Axiom of Choice, which guarantees the existence of maximal elements in certain partially ordered sets.
• C. Tychonoff theorem for products of compact spaces
  The Tychonoff theorem for products of compact spaces is a fundamental result in topology stating that any product of compact topological spaces is compact, a statement that is equivalent in strength to the axiom of choice.
• D. Ky Fan’s lemma
  Ky Fan’s lemma is a combinatorial topological result that generalizes Tucker’s lemma and provides conditions guaranteeing the existence of certain balanced or fully labeled simplices in labeled triangulations of spheres or simplices.
• E. Banach–Alaoglu theorem
  The Banach–Alaoglu theorem is a fundamental result in functional analysis stating that the closed unit ball in the dual of a normed space is compact in the weak-* topology.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Ultrafilter lemma
Target entity description: The ultrafilter lemma is a set-theoretic principle weaker than the full Axiom of Choice that guarantees every filter can be extended to an ultrafilter and underlies several key results in topology and analysis.

• A. Hausdorff maximal principle
  The Hausdorff maximal principle is a foundational result in set theory and order theory stating that every partially ordered set contains a maximal totally ordered subset (a maximal chain), and it is equivalent to the axiom of choice.
• B. Kuratowski–Zorn lemma (attribution as Zorn’s lemma variant)
  The Kuratowski–Zorn lemma is a fundamental result in set theory and order theory, equivalent to the Axiom of Choice, which guarantees the existence of maximal elements in certain partially ordered sets.
• C. Tychonoff theorem for products of compact spaces
  The Tychonoff theorem for products of compact spaces is a fundamental result in topology stating that any product of compact topological spaces is compact, a statement that is equivalent in strength to the axiom of choice.
• D. Ky Fan’s lemma
  Ky Fan’s lemma is a combinatorial topological result that generalizes Tucker’s lemma and provides conditions guaranteeing the existence of certain balanced or fully labeled simplices in labeled triangulations of spheres or simplices.
• E. Banach–Alaoglu theorem
  The Banach–Alaoglu theorem is a fundamental result in functional analysis stating that the closed unit ball in the dual of a normed space is compact in the weak-* topology.
• F. None of above. chosen