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