Definition:Diameter of Subset of Metric Space

This page is about Diameter in the context of Metric Spaces. For other uses, see Diameter.

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.

Sources

• 1953: Walter Rudin: Principles of Mathematical Analysis ... (previous): $3.9$
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Definition $2.2.13$
• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Complete Metric Spaces
• 1984: Gerald B. Folland: Real Analysis: Modern Techniques and their Applications : $\S \text P.6$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): diameter: 3.
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): diameter of a set (metric space)
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): diameter of a set (metric space)