# Kuratowski's Lemma

## Theorem

### Formulation 1

Let $\struct {S, \preceq}, S \ne \O$ be a non-empty ordered set.

Then every chain in $S$ is the subset of some maximal chain.

### Formulation 2

Let $S$ be a set of sets which is closed under chain unions.

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

## Also known as

**Kuratowski's Lemma** is also known as **Kuratowski's Maximal Principle**.

## Kuratowski's Lemma and Axiom of Choice

### Axiom of Choice implies Kuratowski's Lemma

Let the Axiom of Choice be accepted.

Then Kuratowski's Lemma holds.

### Kuratowski's Lemma implies Axiom of Choice

Kuratowski's Lemma implies Axiom of Choice

## Also see

- Results about
**Kuratowski's lemma**can be found**here**.

## Source of Name

This entry was named for Kazimierz Kuratowski.

## Historical Note

Kazimierz Kuratowski published what is now known as **Kuratowski's Lemma** in $1922$, thinking it little more than a corollary of Hausdorff's Maximal Principle.

In $1935$, Max August Zorn published his own equivalent, now known as Zorn's Lemma, acknowledging Kuratowski's earlier work.

This later version became the more famous one.