Definition:Closed Ball/Metric Space

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Definition

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

Let $a \in A$.

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


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

$\map { {B_\epsilon}^-} a := \set {x \in A: \map d {x, a} \le \epsilon}$

where $B^-$ recalls the notation of topological closure.


If it is necessary to show the metric itself, then the notation $\map { {B_\epsilon}^-} {a; d}$ can be used.


Radius

The value $\epsilon$ is referred to as the radius of $\map { {B_\epsilon}^-} a$.


Center

In $\map {B^-_\epsilon} a$, the value $a$ is referred to as the center of the closed $\epsilon$-ball.


Also denoted as

The notation $\map {B^-} {a; \epsilon}$ can be found for $\map { {B_\epsilon}^-} a$, particularly when $\epsilon$ is a more complicated expression than a constant.

Similarly, some sources allow $\map { {B_d}^-} {a; \epsilon}$ to be used for $\map { {B_\epsilon}^-} {a; d}$.

It needs to be noticed that the two styles of notation allow a potential source of confusion, so it is important to be certain which one is meant.


Also see

  • Results about closed balls can be found here.


Sources