# Definition:Open Ball/Radius

< Definition:Open Ball(Redirected from Definition:Radius of Open Ball)

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

## Contents

## 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.

## Caution

It should be noted that the **radius** is not intrinsic to the open ball, so that *the radius of an open ball* is ambiguous.

## 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.

## Sources

- 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