Definition:Bounded Metric Space/Definition 1

Definition

Let $M = \left({A, d}\right)$ be a metric space.

Let $M' = \left({B, d_B}\right)$ be a subspace of $M$.

$M'$ is bounded (in $M$) if and only if:

$\exists a \in A, K \in \R: \forall x \in B: d \left({x, a}\right) \le K$

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

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2.2$: Examples: Definitions $2.2.12$