Bounds

Lower Bounds

Definition: Lower Bound

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

An element $l\in S$ is called a lower bound of $T$ iff

$$l\le t\forall t\in T$$

Infimum

Definition: Infimum

Let $L$ be the set of all lower bounds of $T$.

The infimum $\inf (T)$ of $T$ is its smallest lower bound.

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

Theorem: Uniqueness of the Infimum

The infimum of $T$ is unique.

PROOF

TODO

Upper Bounds

Definition: Upper Bound

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

An element $u\in S$ is called an upper bound of $T$ iff

$$t\le u\forall t\in T$$

Supremum

Definition: Supremum

Let $U$ be the set of all upper bounds of $T$.

The supremum $\sup (T)$ of $T$ is its largest upper bound.

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

Theorem: Uniqueness of the Supremum

The supremum of $T$ is unique.

PROOF

TODO

Boundedness

Definition: Bounded Sets

A partially ordered set is:

• bounded above if it has an upper bound;
• bounded below if it has a lower bound;
• bounded if it has both an upper bound and a lower bound.