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

• Kuratowski's Lemma implies Zorn's Lemma

• Zorn's Lemma
• Kneser's Lemma
• Tukey's Lemma
• Hausdorff's Maximal Principle

• 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.