# Definition:Bounded Below Set

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*This page is about ordered sets which are bounded below. For other uses, see Definition:Bounded Below.*

## Definition

Let $\left({S, \preceq}\right)$ 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 $\left({S, \preceq}\right)$ is the set of real numbers under the usual ordering $\left({\R, \le}\right)$:

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 $\left({S, \preceq}\right)$ be an ordered set.

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

## Also see

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 14$ - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations