Definition:Bounded Set
This page is about Bounded in the context of Ordered Set. For other uses, see Bounded.
Definition
Let $\struct {S, \preceq}$ be an ordered set.
Let $T \subseteq S$ be both bounded below and bounded above in $S$.
Then $T$ is bounded 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 $T \subseteq \R$ be both bounded below and bounded above in $\R$.
Then $T$ is bounded in $\R$.
Unbounded Set
Let $\struct {S, \preceq}$ be an ordered set.
A subset $T \subseteq S$ is unbounded (in $S$) if and only if it is not bounded.
Also known as
A bounded set can also be referred to as a bounded ordered set.
However, it is unnecessary to refer specifically to the ordered nature of such a set, as boundedness cannot be applied to a set which has no ordering.
The term bounded poset can also be found.
Some sources use the term order-bounded set.
Also see
- Results about bounded sets can be found here.
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Orderings
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings