Appendix A - Zorn's lemma and the well-ordering principle
Published online by Cambridge University Press: 05 May 2013
Summary
Zorn's lemma
We show that Zorn's lemma is a consequence of the axiom of choice.
Theorem A.1.1Assume the axiom of choice. Suppose that (X, ≤) is a non-empty partially ordered set with the property that every non-empty chain (totally ordered subset) of X has an upper bound. Then there exists a maximal element in X.
Proof We need a few more definitions. Suppose that A is a subset of a partially ordered set (X, ≤) and that x ∈ X. x is a strict upper bound for A if a < x for all a ∈ A. A totally ordered set (S, ≤)is well-ordered if every non-empty subset of S has a least element. A subset D of a totally ordered set (S, ≤)is an initial segment of S if whenever x ∈ S, d ∈ D and x ≤ d then x ∈ D.
We break the proof into a sequence of lemmas and corollaries. If A is a subset of X, let A be the set of strict upper bounds of A. If A = ∅, then A = X.
Let C be the set of chains in X.
Lemma A.1.2Suppose that there exists C ∈C for which C = ∅. Then C has a unique upper bound, which is a maximal element of X.
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- Information
- A Course in Mathematical Analysis , pp. 291 - 294Publisher: Cambridge University PressPrint publication year: 2013