Definition:Bounded Metric Space/Definition 3

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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