# Definition:Bounded Below Set

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

## Definition

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

A subset $T \subseteq S$ is **bounded below (in $S$)** if and only if $T$ admits a lower 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 below (in $\R$)** if and only if $T$ admits a lower bound (in $\R$).

## Unbounded Below

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

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

## Also see

- Results about
**bounded below 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