Bounded set

A bounded set in a metric space $(X,d)$ is a subset $A\subseteq X$ for which 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 $x_0$.

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