502 – Equivalents of the axiom of choice

November 11, 2009

The goal of this note is to show the following result:

Theorem 1 The following statements are equivalent in {{\sf ZF}:}

  1. The axiom of choice: Every set can be well-ordered.
  2. Every collection of nonempty set admits a choice function, i.e., if {x\ne\emptyset} for all {x\in I,} then there is {f:I\rightarrow\bigcup I} such that {f(x)\in x} for all {x\in I.}
  3. Zorn’s lemma: If {(P,\le)} is a partially ordered set with the property that every chain has an upper bound, then {P} has maximal elements.
  4. Any family of pairwise disjoint nonempty sets admits a selector, i.e., a set {S} such that {|S\cap x|=1} for all {x} in the family.
  5. Any set is a well-ordered union of finite sets of bounded size, i.e., for every set {x} there is a natural {m,} an ordinal {\alpha,} and a function {f:\alpha\rightarrow{\mathcal P}(x)} such that {|f(\beta)|\le m} for all {\beta<\alpha,} and {\bigcup_{\beta<\alpha}f(\beta)=x.}
  6. Tychonoff’s theorem: The topological product of compact spaces is compact.
  7. Every vector space (over any field) admits a basis.

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