In mathematics, the Hausdorff maximal principle is an alternate and earlier formulation of Zorn's lemma proved by

Felix Hausdorff Felix Hausdorff ( , ; November 8, 1868 – January 26, 1942) was a 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, and f ...

in 1914 (Moore 1982:168). It states that in any

partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...

, every

totally ordered In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexiv ...

subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...

is contained in a maximal totally ordered subset. The Hausdorff maximal principle is one of many statements equivalent to the

axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...

over ZF (

Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such a ...

without the axiom of choice). 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

, every

is contained in a maximal totally ordered subset (a totally ordered subset that, if enlarged in any way, does not remain totally ordered). In general, there may be many maximal totally ordered subsets containing a given totally ordered subset. An equivalent form of the Hausdorff maximal principle is that in every partially ordered set there exists a maximal totally ordered subset. To prove that this statement follows from the original form, let ''A'' be a partially ordered set. Then

\varnothing

is a totally ordered subset of ''A'', hence there exists a maximal totally ordered subset containing

\varnothing

, hence 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

\subseteq

, therefore it contains a maximal totally ordered subset ''P''. Then the set

P

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

Examples

If ''A'' is any collection of sets, the relation "is a proper subset of" is a

strict partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binar ...

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. If (x₀, y₀) and (x₁, y₁) are two points of the plane ℝ², define (x₀, y₀) < (x₁, y₁) if y₀ = y₁ and x₀ < x₁. This is a partial ordering of ℝ² under which two points are comparable only if they lie on the same horizontal line. The maximal totally ordered sets are horizontal lines in ℝ².

References

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

James Munkres James Raymond Munkres (born August 18, 1930) is a Professor Emeritus of mathematics at MIT and the author of several texts in the area of topology, including ''Topology'' (an undergraduate-level text), ''Analysis on Manifolds'', ''Elements of Alge ...

(2000), ''Topology'', Pearson. {{Order theory Axiom of choice Mathematical principles Order theory