# Definition:Bounded 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 Set

Let $\struct {S, \preceq}$ 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 set** can also be referred to as a **bounded ordered set**.

However, it is unnecessary to refer specifically to the ordered nature of such a set, as boundedness cannot be applied to a set which has no ordering.

The term **bounded poset** can also be found.

Some sources use the term **order-bounded set**.

## Also see

- Results about
**bounded sets**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