# Definition:Bounded Above Set

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

## Definition

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

A subset $T \subseteq S$ is bounded above (in $S$) if and only if $T$ admits an upper bound (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 $\R$ be the set of real numbers.

A subset $T \subseteq \R$ is bounded above (in $\R$) if and only if $T$ admits an upper bound (in $\R$).

### Unbounded Above

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

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