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:
\(\ds \forall x \in A: \, \) | \(\ds x\) | \(\ne\) | \(\ds a\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map d {x, a}\) | \(\ge\) | \(\ds \epsilon\) | Definition of Standard Discrete Metric | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(\notin\) | \(\ds \map {B_\epsilon} {a; d}\) | Definition of Open $\epsilon$-Ball | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {B_\epsilon} {a; d}\) | \(=\) | \(\ds \set a\) |
$\Box$
Let $\epsilon \in \R_{>0}: \epsilon > 1$.
Then:
\(\ds \forall x \in A: \, \) | \(\ds \map d {x, a}\) | \(>\) | \(\ds \epsilon\) | Definition of Standard Discrete Metric | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(\in\) | \(\ds \map {B_\epsilon} {a; d}\) | Definition of Open $\epsilon$-Ball | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {B_\epsilon} {a; d}\) | \(=\) | \(\ds A\) |
Hence the result.
$\blacksquare$
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.3$: Open sets in metric spaces: Example $2.3.4$