Definition:Closed Ball/Metric Space
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
- 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