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:
 * $\map {S_\epsilon} {a; d} = \map { {B_\epsilon}^-} {a; d} \setminus \map {B_\epsilon} {a; d}$