Definition:Bounded Ordered Set

This page is about Bounded in the context of Ordered Set. For other uses, see Bounded.

Definition

Let $\struct {S, \preceq}$ be an ordered set.

Let $T \subseteq S$ be both bounded below and bounded above in $S$.

Then $T$ is bounded in $S$.

Subset of Real Numbers

The concept is usually encountered where $\struct {S, \preceq}$ is the set of real numbers under the usual ordering $\struct {\R, \le}$:

Let $T \subseteq \R$ be both bounded below and bounded above in $\R$.

Then $T$ is bounded in $\R$.

Unbounded Ordered Set

Let $\left({S, \preceq}\right)$ be an ordered set.

A subset $T \subseteq S$ is unbounded (in $S$) if and only if it is not bounded.

Also known as

A bounded ordered set can also be referred to as a bounded poset.

Some sources use the term order-bounded.

Also see

• Results about boundedness can be found here.

Sources

• 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Orderings
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings