# Hausdorff's Maximal Principle

(Redirected from Hausdorff Maximal Principle)

## Theorem

### Formulation 1

Let $\struct {\PP, \preceq}$ be a non-empty partially ordered set.

Then there exists a maximal chain in $\PP$.

### Formulation 2

Let $A$ be a non-empty set of sets.

Let $S$ be the set of all chain of sets of $A$ (ordered under the subset relation).

Then every element of $S$ is a subset of a maximal element of $S$ under the subset relation.

## Hausdorff's Maximal Principle and Axiom of Choice

### Axiom of Choice implies Hausdorff's Maximal Principle

Let the Axiom of Choice be accepted.

Then Hausdorff's Maximal Principle holds.

### Hausdorff's Maximal Principle implies Axiom of Choice

Let Hausdorff's Maximal Principle be accepted.

Then the Axiom of Choice holds.

## Also known as

Hausdorff's Maximal Principle is also known as the Hausdorff Maximal Principle.

Some sources call it the Hausdorff Maximality Principle or the Hausdorff Maximality Theorem.

## Also see

• Results about Hausdorff's maximal principle can be found here.

## Source of Name

This entry was named for Felix Hausdorff.