Kuratowski's Lemma/Formulation 1

Theorem

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.

Also known as

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

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.

Sources

• 1922: Kazimierz Kuratowski: Une méthode d'élimination des nombres transfinis des raisonnements mathématiques (Fund. Math. Vol. 3: pp. 76 – 108)
• 1935: Max August Zorn‎: A remark on method in transfinite algebra (Bull. Amer. Math. Soc. Vol. 41: pp. 667 – 670)
• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: Notes

• This article incorporates material from Equivalence of Kuratowski’s lemma and Zorn’s lemma on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.