Definition:Bounded Metric Space/Also defined as

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Bounded Metric Space: Also defined as

Some sources place no emphasis on the fact that the subset $B$ of the underlying set $A$ of $M$ is in fact itself a subspace of $M'$, and merely refer to a bounded set.

This, however, glosses over the facts that:

$\text{(a)}$: from Subspace of Metric Space is Metric Space, any such subset is also a metric space by dint of the induced metric $d_B$
$\text{(b)}$: without reference to such a metric, boundedness is not defined.

Hence $\mathsf{Pr} \infty \mathsf{fWiki}$ strives to ensure that boundedness is consistently defined in the context of a metric space, and not just a subset.