# 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:

 $\, \displaystyle \forall x \in A: \,$ $\displaystyle x$ $\ne$ $\displaystyle a$ $\displaystyle \leadsto \ \$ $\displaystyle \map d {x, a}$ $\ge$ $\displaystyle \epsilon$ Definition of Standard Discrete Metric $\displaystyle \leadsto \ \$ $\displaystyle x$ $\notin$ $\displaystyle \map {B_\epsilon} {a; d}$ Definition of Open $\epsilon$-Ball $\displaystyle \leadsto \ \$ $\displaystyle \map {B_\epsilon} {a; d}$ $=$ $\displaystyle \set a$

$\Box$

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

Then:

 $\, \displaystyle \forall x \in A: \,$ $\displaystyle \map d {x, a}$ $>$ $\displaystyle \epsilon$ Definition of Standard Discrete Metric $\displaystyle \leadsto \ \$ $\displaystyle x$ $\in$ $\displaystyle \map {B_\epsilon} {a; d}$ Definition of Open $\epsilon$-Ball $\displaystyle \leadsto \ \$ $\displaystyle \map {B_\epsilon} {a; d}$ $=$ $\displaystyle A$

Hence the result.

$\blacksquare$