Let $S$ be a (non-empty) ordered set.
Let $C$ be a chain in $S$.
Let $P$ be the set of all chains that are supersets of $C$.
Let $\CC$ be a chain in $\powerset P$ (partially ordered by set-inclusion).
Define $C' = \bigcup \CC$.
Note that the elements of $P$ are chains on $\paren S$, so the elements of $\CC$ are also chains in $S$, as $\CC$ is a subset of $P$.
Thus $\bigcup \CC$ contains elements in $S$, so:
- $C' \subseteq S$.
First, note that $C'$ is a chain in $S$.
Let $x, y \in C'$, which means $x \in X$ and $y \in Y$ for some $X, Y \in \CC$.
However, as $\CC$ is a chain in $\powerset P$, that means either $X \subseteq Y$ or $Y \subseteq X$.
So $x$ and $y$ belong to the same chain in $S$.
Thus either $x \le y$ or $y \le x$.
Thus $C'$ is a chain on $S$.
Now let $x \in C$.
- $\forall A \in P: x \in A$
Then because $\CC \subseteq P$:
- $\forall A \in \CC: x \in A$
- $x \in \bigcup \CC$
and so $C \subseteq C'$
- $C' \in P$
Now, note $C'$ is an upper bound on $\CC$.
To prove this consider $x \in D \in \CC$.
- $x \in \bigcup \CC = C'$
- $D\subseteq C'$
The chain in $P$ was arbitrary, so every chain in $P$ has an upper bound.
Thus, by Zorn's Lemma, $P$ has a maximal element.
This must be a maximal chain containing $C$.
One can also prove that Zorn's lemma follows from Kuratowski's Lemma, which shows that they are equivalent statements. Thus, this is another statement equivalent to the Axiom of Choice.
Source of Name
This entry was named for Kazimierz Kuratowski.
This later version became the more famous one.
- 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 incorporates material from Equivalence of Kuratowski’s lemma and Zorn’s lemma on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.