\begin{definition}[Definition:Closed Ball/Metric Space]
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.
\end{definition}