Definition:Bounded Below Set

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 $S$).

The concept is usually encountered where $\left({S, \preceq}\right)$ is the set of real numbers under the usual ordering $\left({\R, \le}\right)$:

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

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.