## You are here

HomeHausdorff's maximum principle

## Primary tabs

# Hausdorff’s maximum principle

Theorem Let $X$ be a partially ordered set. Then there exists a maximal totally ordered subset of $X$.

The Hausdorff’s maximum principle is one of the many theorems equivalent to the axiom of choice. The below proof uses Zorn’s lemma, which is also equivalent to the axiom of choice.

###### Proof.

Let $S$ be the set of all totally ordered subsets of $X$. $S$ is not empty, since the empty set is an element of $S$. Partial order $S$ by inclusion. Let $\tau$ be a chain (of elements) in $S$. Being each totally ordered, the union of all these elements of $\tau$ is again a totally ordered subset of $X$, and hence an element of $S$, as is easily verified. This shows that $S$, ordered by inclusion, is inductive. The result now follows from Zorn’s lemma. ∎

## Mathematics Subject Classification

03E25*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new question: numerical method (implicit) for nonlinear pde by roozbe

new question: Harshad Number by pspss

Sep 14

new problem: Geometry by parag

Aug 24

new question: Scheduling Algorithm by ncovella

new question: Scheduling Algorithm by ncovella

## Corrections

proof supplied by Wkbj79 ✓

(small) error by asteroid ✓

Contains own proof by rm50 ✓

Alternate names by flamingspinach ✓