# 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.