Open Ball in Standard Discrete Metric Space

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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.

$\blacksquare$


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