equivalence of Kuratowski’s lemma and Zorn’s lemma


In this entry, we prove the equivalence of Kuratowski’s lemma and Zorn’s lemma, thereby establishing the equivalence of Kuratowski’s lemma and the axiom of choiceMathworldPlanetmath. The proof, of course, only uses ZF axioms.

Proposition 1.

Kuratowski’s lemma implies Zorn’s lemma.

Proof.

Suppose P is a poset such that every chain has an upper bound. Let C be a chain in P. By Kuratowski’s lemma, C can be extended to a maximal chain C, which, by assumptionPlanetmathPlanetmath, has an upper bound aP. Suppose for some bP, ab. If ba, then ba, or bC, which means C{b} is a chain extending C, thus extending C. But this means that C is not maximal, a contradictionMathworldPlanetmathPlanetmath. This shows that b=a, or that a is a maximal elementMathworldPlanetmath in P, proving Zorn’s lemma. ∎

Proposition 2.

Zorn’s lemma implies Kuratowski’s lemma.

Proof.

Suppose P is a poset and C a chain in P. We assume that P. Let 𝒫 be the set of all chains in P extending C. Partially order 𝒫 by inclusion so that 𝒫 is a poset. Let 𝒞 be a chain in 𝒫. Let C=𝒞. We want to prove that C is an upper bound of 𝒞 in 𝒫.

  1. 1.

    C is a chain in P. If x,yC, then xD and yE for some D,E𝒞. Since 𝒞 is a chain in 𝒫, DE or ED, which implies that x,y belong to the same chain (either D or E) in P. So xy or yx. This shows that C is a chain in P.

  2. 2.

    C is in 𝒫. If aC, then aD for every D𝒞, and therefore aC, showing that C extends C, or C𝒫.

  3. 3.

    C is an upper bound of 𝒞. Pick any xD, for an arbitrary D𝒞. Then x𝒞=C, so DC. Since D is arbitrary, C is an upper bound of 𝒞.

By Zorn’s lemma, 𝒫 has a maximal element M. Then M is a maximal chain in P extending C, for if there is a chain N in P such that MN, then N𝒫 and M would no longer be maximal. ∎

Title equivalence of Kuratowski’s lemma and Zorn’s lemma
Canonical name EquivalenceOfKuratowskisLemmaAndZornsLemma
Date of creation 2013-03-22 18:44:23
Last modified on 2013-03-22 18:44:23
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 6
Author CWoo (3771)
Entry type Proof
Classification msc 03E25