Definition:Bounded Metric Space/Definition 3

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 $M$) if and only if:

$\exists x \in A, \epsilon \in \R_{>0}: B \subseteq \map {B_\epsilon} x$

where $\map {B_\epsilon} x$ is the open $\epsilon$-ball of $x$.

That is, $M'$ can be fitted inside an open ball.

Also see

• Equivalence of Definitions of Bounded Metric Space

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: Compactness