Definition:Bounded Metric Space/Definition 1

Definition
Let $M = \struct {A, d}$ be a metric space.

Let $M' = \struct {B, d_B}$ be a subspace of $M$.

$M'$ is bounded (in $M$) :
 * $\exists a \in A, K \in \R: \forall x \in B: \map {d} {x, a} \le K$

That is, there exists an element of $A$ within a finite distance of all elements of $B$.

Also see

 * Equivalence of Definitions of Bounded Metric Space