Definition:Diameter of Bounded Metric Subspace

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Let $M = \struct {A, d}$ be a metric space.

Let $S \subseteq A$ be bounded in $M$.

Then the diameter of $S$ is defined as:

$\map {\operatorname {diam} } S := \sup \set {\map d {x, y}: x, y \in S}$

That is, by the definition of the supremum, $\map {\operatorname {diam} } S$ is the smallest real number $D$ such that any two points of $S$ are at most a distance $D$ apart.