Definition:Sphere/Metric Space

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Definition

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

Let $a \in A$.

Let $\epsilon \in \R_{>0}$ be a strictly positive real number.


The $\epsilon$-sphere of $a$ in $M$ is defined as:

$\map {S_\epsilon} a = \set {x \in A: \map d {x, a} = \epsilon}$


Radius

In $S_\epsilon \paren{a}$, the value $\epsilon$ is referred to as the radius of the $\epsilon$-sphere.


Center

In $S_\epsilon \paren{a}$, the value $a$ is referred to as the center of the $\epsilon$-sphere.


Sources

This article incorporates material from sphere (metric space) on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.