Definition:Open Ball/Radius

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This page is about the radius of an open ball. For other uses, see Definition:Radius.

Definition

Let $M = \left({A, d}\right)$ be a metric space or pseudometric space.

Let $a \in A$.

Let $B_\epsilon \left({a}\right)$ be the open $\epsilon$-ball of $a$.


In $B_\epsilon \left({a}\right)$, 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.


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.


Sources