Sphere is Set Difference of Closed Ball with Open Ball

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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

$\blacksquare$