Hausdorff Maximal Principle - Statement

Statement

The Hausdorff maximal principle states that, in any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset. Here a maximal totally-ordered subset is one that, if enlarged in any way, does not remain totally ordered. The maximal set produced by the principle is not unique, in general; there may be many maximal totally ordered subsets containing a given totally ordered subset.

An equivalent form of the principle is that in every partially ordered set there exists a maximal totally ordered subset.

To prove that it follows from the original form, let A be a poset. Then is a totally ordered subset of A, hence there exists a maximal totally ordered subset containing, in particular A contains a maximal totally ordered subset.

For the converse direction, let A be a partially ordered set and T a totally ordered subset of A. Then

is partially ordered by set inclusion, therefore it contains a maximal totally ordered subset P. Then the set satisfies the desired properties.

The proof that the Hausdorff maximal principle is equivalent to Zorn's lemma is very similar to this proof.

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