Diameter

The supremum of distances between pairs of points in a set.

Diameter

Let $(X,d)$ be a metric space and let $A\subseteq X$ . The diameter of $A$ is

\operatorname{diam}(A):=\sup\{d(x,y): x,y\in A\}\in[0,\infty].

(If the set of distances is unbounded, the supremum is $+\infty$ .)

Diameter measures the “size” of a set in terms of its maximal pairwise separation. It is used in compactness and completeness arguments (e.g., nested closed sets with diameters $\to 0$ ).

Examples:

In $\mathbb{R}$ , $\operatorname{diam}([a,b])=b-a$ .
In $\mathbb{R}^2$ , the diameter of the closed unit disk $\overline{B}(0,1)$ is $2$ .
In $\mathbb{R}$ , $\operatorname{diam}(\mathbb{Z})=+\infty$ .