# Kuratowski's Lemma

## Theorem

Let $\left({S, \preceq}\right), S \ne \varnothing$ be a non-empty ordered set.

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

## Source of Name

This entry was named for Kazimierz Kuratowski.

He published this in 1922, thinking it little more than a corollary of the Hausdorff Maximal Principle.

In 1935, Max August Zorn published a similar version, acknowledging Kuratowski's earlier work. This later version became the more famous one.