# Hausdorff's Maximal Principle

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## 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.