Definition:Bounded Metric Space/Definition 2
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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:
- $\exists K \in \R: \forall x, y \in M': \map {d_B} {x, y} \le K$
That is, there exists a finite distance such that all pairs of elements of $B$ are within that distance.
Also see
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Complete Metric Spaces