Bounded set - Knowlpedia

A bounded set in a metric space $(X,d)$ is a subset $A\subseteq X$ such that either $A=\varnothing$ , or there exist $x_0\in X$ and $R>0$ such that

A\subseteq B_d(x_0,R),

where $B_d(x_0,R)$ is the open ball of radius $R$ around $x_0$ .

Equivalent characterizations

Equivalently, $A$ is bounded if and only if its diameter is finite.

Remarks

Boundedness is a basic size condition that appears in notions such as total boundedness.

Examples

Any (open or closed) ball of finite radius is bounded.
In $(\mathbb{R},|\cdot|)$ , the interval $(0,1)$ is bounded, while $(0,\infty)$ is not.