Definition:Diameter of Bounded Metric Subspace

From ProofWiki
Jump to navigation Jump to search

Definition

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.


Sources