# Kuratowski's Lemma

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## Contents

## 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.

## Proof

## Also see

## 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.

## Sources

- 1922: Kazimierz Kuratowski:
*Une méthode d'élimination des nombres transfinis des raisonnements mathématiques*(*Fund. Math.***Vol. 3**: 76 – 108) - 1935: Max August Zorn:
*A remark on method in transfinite algebra*(*Bull. Amer. Math. Soc.***Vol. 41**: 667 – 670)

- 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): $\S 1.1$: Definition $1$