Kuratowski's Lemma

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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