# Definition:Diameter of Subset of Metric Space

## Definition

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

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

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

$\map \diam S := \begin {cases} \sup \set {\map d {x, y}: x, y \in S} & : \text {if this quantity is finite} \\ + \infty & : \text {otherwise} \end {cases}$

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

If $d: S^2 \to \R$ does not admit a supremum, then $\map \diam S$ is infinite.