Definition:Open Ball/Normed Division Ring

Definition
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.

The open $\epsilon$-ball of $a$ in $\struct{R, \norm{\,\cdot\,}}$ is defined as:


 * $B_\epsilon \paren{a} = \set {x \in R: \norm{x - a} \lt \epsilon}$

Let $d$ be the metric induced by the norm $\norm{\,\cdot\,}$.

By the definition of the metric induced by the norm, the open $\epsilon$-ball of $a$ in $\struct{R, \norm{\,\cdot\,}}$ is the open $\epsilon$-ball of $a$ in $\struct{R, d}$

If it is necessary to show the norm itself, then the notation $B_\epsilon \paren{a; \norm{\,\cdot\,}}$ can be used.