Hausdorff maximal principle

# Hausdorff maximal principle

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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Hausdorff maximal principle is an alternate and earlier formulation of Zorn's lemma
Zorn's lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory that states:Suppose a partially ordered set P has the property that every chain has an upper bound in P...

proved by Felix Hausdorff
Felix Hausdorff
Felix Hausdorff was a Jewish German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, function theory, and functional analysis.-Life:Hausdorff studied at the University of Leipzig,...

in 1914 (Moore 1982:168). It states that in any partially ordered set, every totally ordered
Total order
In set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...

subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

is contained in a maximal totally ordered subset.

The Hausdorff maximal principle is one of many statements equivalent to the axiom of choice over Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory
In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes...

. The principle is also called the Hausdorff maximality theorem or the Kuratowski lemma (Kelley 1955:33).

## Statement

The Hausdorff maximal principle states that, in any partially ordered set, every totally ordered
Total order
In set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...

subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

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.

## Reference

• John Kelley (1955), General topology, Von Nostrand.
• Gregory Moore (1982), Zermelo's axiom of choice, Springer.