# Definition:Diameter of Subset of Metric Space

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## Definition

Let $M = \left({A, d}\right)$ be a metric space.

Let $S \subseteq A$ be subset of $A$.

Then the diameter of $S$ is the extended real number defined by:

$\operatorname {diam} \left({S}\right) := \begin{cases} \sup \left\{ {d \left({x, y}\right): x, y \in S }\right\}, \ \text{if this quantity is finite} \\ +\infty, \ \text{otherwise} \end{cases}$

In the finite case, this is, by the definition of the supremum, the smallest real number $D$ such that any two points of $S$ are at most a distance $D$ apart.