Boundedness

Definition: Boundedness

Let $S$ be a partially ordered set and let $T$ be subset of $S$.

We say that $T$ is:

• bounded below if there exists some $l\in S$ such that $l\le t$ for all $t\in T$. Any such $l$ is called a lower bound of $T$;
• bounded above if there exists some $u\in S$ such that $t\le u$ for all $t\in T$. Any such $u$ is called an upper bound of $T$;
• bounded if it is both bounded above and bounded below.

Definition: Infimum

Let $S$ be a partially ordered set, let $T\subseteq S$ be bounded below and let $L$ be the set of all lower bounds of $T$.

The infimum of $T$ is the smallest lower bound of $T$:

$$\inf (T)\le l\forall l\in L$$

Definition: Supremum

Let $S$ be a partially ordered set, let $T\subseteq S$ be bounded above and let $U$ be the set of all upper bounds of $T$.

The supremum of $T$ is the greatest upper bound of $T$:

$$u\le \sup (T)\forall u\in U$$