Kuratowski's Lemma/Formulation 2

Theorem

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.

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

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text {II}$ -- Maximal principles: $\S 5$ Maximal principles