Open Ball in Standard Discrete Metric Space

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

Let $d$ be the standard discrete metric on $M$.

Let $a \in A$.

Let $\map {B_\epsilon} {a; d}$ be an open $\epsilon$-ball of $a$ in $M$.

Then:
 * $\map {B_\epsilon} {a; d} = \begin {cases}

\set a & : \epsilon \le 1 \\ A & : \epsilon > 1 \end {cases}$

Proof
Let $\epsilon \in \R_{>0}: \epsilon \le 1$.

Then:

Let $\epsilon \in \R_{>0}: \epsilon > 1$.

Then:

Hence the result.