Definition:Bounded Set

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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