Hausdorff paradox

Target entity: Hausdorff paradox
Context triple: [Banach–Tarski paradox, relatedTo, Hausdorff paradox]

• A. Banach–Tarski paradox
  The Banach–Tarski paradox is a theorem in set-theoretic geometry stating that a solid ball in 3‑dimensional space can be decomposed into finitely many non-measurable pieces and reassembled into two identical copies of the original ball, highlighting counterintuitive consequences of the axiom of choice.
• B. Mazurkiewicz–Sierpiński paradox
  The Mazurkiewicz–Sierpiński paradox is a result in set-theoretic geometry showing that a sphere can be decomposed and reassembled in a counterintuitive way, illustrating the existence of paradoxical decompositions similar to the Banach–Tarski paradox.
• C. Steinhaus theorem
  The Steinhaus theorem is a fundamental result in measure theory stating that the difference set of any subset of the real numbers with positive Lebesgue measure contains an open interval around zero.
• D. Hausdorff
  Hausdorff is a topological separation property requiring that any two distinct points in a space can be enclosed in disjoint open sets.
• E. Sierpiński set
  The Sierpiński set is a subset of the real numbers with the property that it intersects every uncountable closed subset of the reals in only countably many points, illustrating extreme pathological behavior in set theory and real analysis.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Hausdorff paradox
Target entity description: The Hausdorff paradox is a result in set-theoretic geometry showing that, using the axiom of choice, a sphere can be decomposed into finitely many disjoint pieces that can be reassembled into a set not congruent to the original, illustrating the existence of non-measurable sets and paving the way for the Banach–Tarski paradox.

• A. Banach–Tarski paradox
  The Banach–Tarski paradox is a theorem in set-theoretic geometry stating that a solid ball in 3‑dimensional space can be decomposed into finitely many non-measurable pieces and reassembled into two identical copies of the original ball, highlighting counterintuitive consequences of the axiom of choice.
• B. Mazurkiewicz–Sierpiński paradox
  The Mazurkiewicz–Sierpiński paradox is a result in set-theoretic geometry showing that a sphere can be decomposed and reassembled in a counterintuitive way, illustrating the existence of paradoxical decompositions similar to the Banach–Tarski paradox.
• C. Steinhaus theorem
  The Steinhaus theorem is a fundamental result in measure theory stating that the difference set of any subset of the real numbers with positive Lebesgue measure contains an open interval around zero.
• D. Hausdorff
  Hausdorff is a topological separation property requiring that any two distinct points in a space can be enclosed in disjoint open sets.
• E. Sierpiński set
  The Sierpiński set is a subset of the real numbers with the property that it intersects every uncountable closed subset of the reals in only countably many points, illustrating extreme pathological behavior in set theory and real analysis.
• F. None of above. chosen