Equivalent Forms of Zorn's Lemma
Zorn's lemma is equivalent (in ZF) to three main results:
- Hausdorff maximal principle
- Axiom of choice
- Well-ordering theorem.
Moreover, Zorn's lemma (or one of its equivalent forms) implies some major results in other mathematical areas. For example,
- Banach's extension theorem which is used to prove one of the most fundamental results in functional analysis, the Hahn–Banach theorem
- Every vector space has a Hamel basis, a result from linear algebra (to which it is equivalent)
- Every commutative unital ring has a maximal ideal, a result from ring theory
- Tychonoff's theorem in topology (to which it is also equivalent)
In this sense, we see how Zorn's lemma can be seen as a powerful tool, especially in the sense of unified mathematics.
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