In mathematics, the Hausdorff maximal principle is an alternate and earlier formulation of Zorn's lemma proved by Felix Hausdorff in 1914 (Moore 1982:168).
It states that in any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset, where "maximal" is with respect to set inclusion.
The Hausdorff maximal principle is one of many statements equivalent to the axiom of choice over ZF (Zermelo–Fraenkel set theory without the axiom of choice).
The principle is also called the Hausdorff maximality theorem or the Kuratowski lemma (Kelley 1955:33).
The Hausdorff maximal principle states that, in any partially ordered set
(i.e., a totally ordered subset) is contained in a maximal chain
An equivalent form of the Hausdorff maximal principle is that in every partially ordered set, there exists a maximal chain.
is partially ordered by set inclusion.
The proof that the Hausdorff maximal principle is equivalent to Zorn's lemma is somehow similar to this proof.
Since a union of a totally ordered set of chains is a chain, the hypothesis of Zorn's lemma (every chain has an upper bound) is satisfied for
Conversely, if the maximal principle holds, then
If A is any collection of sets, the relation "is a proper subset of" is a strict partial order on A.
Suppose that A is the collection of all circular regions (interiors of circles) in the plane.
One maximal totally ordered sub-collection of A consists of all circular regions with centers at the origin.
Another maximal totally ordered sub-collection consists of all circular regions bounded by circles tangent from the right to the y-axis at the origin.
under which two points are comparable only if they lie on the same horizontal line.
The maximal totally ordered sets are horizontal lines in
By the Hausdorff maximal principle, we can show every Hilbert space
contains a maximal orthonormal subset
be the set of all orthonormal subsets of the given Hilbert space
, which is partially ordered by set inclusion.
It is nonempty as it contains the empty set and thus by the maximal principle, it contains a maximal chain
We shall show it is a maximal orthonormal subset.
For the purpose of comparison, here is a proof of the same fact by Zorn's lemma.
is also orthonormal by the same argument as above and so is an upper bound of
The idea of the proof is essentially due to Zermelo and is to prove the following weak form of Zorn's lemma, from the axiom of choice.
[2][3] (Zorn's lemma itself also follows from this weak form.)
The maximal principle follows from the above since the set of all chains in
(This is the moment we needed to collapse a set to an element by the axiom of choice to define
The Bourbaki–Witt theorem can also be used to prove the Hausdorff maximal principle: