Definition:Open Ball/Radius

This page is about radius in the context of Open Ball of Metric Space. For other uses, see radius.

Definition

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

Let $a \in A$.

Let $\map {B_\epsilon} a$ be the open $\epsilon$-ball of $a$.

In $\map {B_\epsilon} a$, the value $\epsilon$ is referred to as the radius of the open $\epsilon$-ball.

Linguistic Note

The plural of radius is radii, pronounced ray-dee-eye.

This irregular plural form stems from the Latin origin of the word radius, meaning ray.

The ugly incorrect form radiuses can apparently be found, but rarely in a mathematical context.

The adjectival form radial usually means directed along a radius.

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 4$: Open Balls and Neighborhoods: Definition $4.1$
• 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