Open Ball in Standard Discrete Metric Space

Theorem
Let $M = \left({A, d}\right)$ be a metric space.

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

Let $a \in A$.

Let $B_\epsilon \left({a; d}\right)$ be an open $\epsilon$-ball of $a$ in $M$.

Then:
 * $B_\epsilon \left({a; d}\right) = \begin{cases}

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

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

Then:

Let

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

Then:

Hence the result.