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62 CHAPTER 2. SOME BASIC TOPICS

on X∞ as follows. If x, y are elements of X∞, pick S ∈C such that x, y are both in S. Then if≤S is the order on S, let x≤ y if and only if x≤S y. This definition is well defined becauseof the definition of the order, ≺. Now let U be any nonempty subset of X∞. Then S∩U ̸= /0for some S ∈ C . Because of the definition of ≤, if y ∈ S2 \ S1, Si ∈ C , then x ≤ y for allx ∈ S1. Thus, if y ∈ X∞ \ S then x ≤ y for all x ∈ S and so the smallest element of S∩Uexists and is the smallest element in U . Therefore X∞ is well-ordered. Now suppose thereexists z ∈ X \X∞. Define the following order, ≤1, on X∞∪{z}.

x≤1 y if and only if x≤ y whenever x,y ∈ X∞

x≤1 z whenever x ∈ X∞.

Then let C̃ = {S ∈ C or X∞∪{z}}. Then C̃ is a strictly larger chain than C contradictingmaximality of C . Thus X \X∞ = /0 and this shows X is well-ordered by ≤. ■

With these two lemmas the main result follows.

Theorem 2.8.4 The following are equivalent.

The axiom of choice

The Hausdorff maximal principle

The well-ordering principle.

Proof: It remains to show that the well-ordering principle implies the axiom of choice.Let I be a nonempty set and let Xi be a nonempty set for each i ∈ I. Let X = ∪{Xi : i ∈ I}and well order X . Let f (i) be the smallest element of Xi. Then f ∈∏i∈I Xi. ■

The book by Hewitt and Stromberg [23] has more equivalences.

2.9 Exercises1. Zorn’s lemma says that if you have a nonempty partially ordered set F and every

chain C has an upper bound, then there is a maximal element in F , some x such thatif x≺ y then x = y. Show this is equivalent to the Hausdorff maximal principle.

2. A Hamel basis is a set of vectors B in a vector space X such that every element of Xcan be written in a unique way as a finite linear combination of vectors of B. Showevery vector space has a Hamel basis. In fact, these are not used much outside offinite dimensional settings because it can be shown that in every complete normedlinear space which is not finite dimensional, the Hamel basis must be uncountablebut it is nice to know they exist.