Partially Order Sets and Zorn's Lemma

Definition: Let $A$ be a set and let $\leq$ be a relation on $A$. A Partially Ordered Set is a pair $(A, \leq)$ that satisfies the following three properties: 1) For all $a \in A$, $a \leq a$. 2) For all $a, b \in A$ if $a \leq b$ and $b \leq a$ then $a = b$. 3) For all $a, b, c \in A$ if $a \leq b$ and $b \leq c$ then $a \leq c$.

For example, consider the set $A = \{ 1, 2, 3, 4, 5 \}$ with the standard relation $\leq$. Then $(A, \leq)$ is a partially order set and properties (1), (2), and (3) can easily be verified.

Of course, partially ordered sets can be much more complicated.

Definition: Let $(A, \leq)$ be a partially order set. A Chain in $A$ is a subset $A_0 \subseteq A$ such that for all $a, b \in A$ we have that either $a \leq b$ or $b \leq a$. An Upper Bound for a chain $A_0$ in $A$ is an element $a_0' \in A_0$ such that $a \leq a_0$ for all $a \in A_0$.

Definition: Let $(A, \leq)$ be a partially ordered set. A Maximal Element of $A$ is an element $a^* \in A$ such that $a \leq a^*$ for all $a \in A$.

We now state a very important result known as Zorn's Lemma.

Lemma 1 (Zorn's Lemma): Let $(A, \leq)$ be a partially ordered set. If every chain $A_0$ in $A$ has an upper bound then $A$ has a maximal element.

This lemma will be used later on in various proofs in the theory of first order ODEs.