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.

Also see

• Definition:Upper Bound of Set

• Definition:Bounded Below Set
• Definition:Lower Bound of Set

• Definition:Bounded Set

• Results about bounded above sets can be found here.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings
• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations