Sphere is Set Difference of Closed Ball with Open Ball
Theorem
Let $M = \struct{A, d}$ be a metric space or pseudometric space.
Let $a \in A$.
Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
Let $\map {{B_\epsilon}^-} {a; d}$ denote the closed $\epsilon$-ball of $a$ in $M$.
Let $\map {B_\epsilon} {a; d}$ denote the open $\epsilon$-ball of $a$ in $M$.
Let $\map {S_\epsilon} {a; d}$ denote the $\epsilon$-sphere of $a$ in $M$.
Then:
- $\map {S_\epsilon} {a; d} = \map { {B_\epsilon}^-} {a; d} \setminus \map {B_\epsilon} {a; d}$
Corollary 1
Let $\struct{R, \norm {\,\cdot\,} }$ be a normed division ring.
Let $a \in R$.
Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
Let $\map {{B_\epsilon}^-} {a; \norm {\,\cdot\,} }$ denote the $\epsilon$-closed ball of $a$ in $\struct {R, \norm {\,\cdot\,} }$.
Let $\map {B_\epsilon} {a; \norm {\,\cdot\,} }$ denote the $\epsilon$-open ball of $a$ in $\struct {R, \norm {\,\cdot\,} }$.
Let $\map {S_\epsilon} {a; \norm {\,\cdot\,} }$ denote the $\epsilon$-sphere of $a$ in $\struct {R, \norm {\,\cdot\,} }$.
Then:
- $\map {S_\epsilon} {a; \norm {\,\cdot\,} } = \map { {B_\epsilon}^-} {a; \norm {\,\cdot\,} } \setminus \map {B_\epsilon} {a; \norm {\,\cdot\,} }$
Corollary 2
Let $p$ be a prime number.
Let $\struct{\Q_p,\norm{\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a \in \Q_p$.
Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
Let $\map {{B_\epsilon}^-} a$ denote the $\epsilon$-closed ball of $a$ in $\Q_p$.
Let $\map {B_\epsilon} a$ denote the $\epsilon$-open ball of $a$ in $\Q_p$.
Let $\map {S_\epsilon} a$ denote the $\epsilon$-sphere of $a$ in $\Q_p$.
Then:
- $\map {S_\epsilon} a = \map { {B_\epsilon}^-} a \setminus \map {B_\epsilon} a$
Proof
\(\ds \map {S_\epsilon } a\) | \(=\) | \(\ds \set {x : \map d {x, a} = \epsilon}\) | Definition of Sphere | |||||||||||
\(\ds \) | \(=\) | \(\ds \set {x : \map d {x, a} \le \epsilon} \setminus \set {x : \map d {x, a} < \epsilon }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {x : \map d {x, a} \le \epsilon} \setminus \map {B_\epsilon} a\) | Definition of Open Ball of Metric Space | |||||||||||
\(\ds \) | \(=\) | \(\ds \map { {B_\epsilon}^- } a \setminus \map {B_\epsilon } a\) | Definition of Closed Ball of Metric Space |
$\blacksquare$