Definition:Open Ball/Radius

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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.


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