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
  • 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 is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.