\begin{definition}[Definition:Equivalent Division Ring Norms/Open Unit Ball Equivalent]
Let $R$ be a division ring.
Let $\norm {\,\cdot\,}_1: R \to \R_{\ge 0}$ and $\norm {\,\cdot\,}_2: R \to \R_{\ge 0}$ be norms on $R$.
Let $d_1$ and $d_2$ be the metrics induced by the norms $\norm{\,\cdot\,}_1$ and $\norm{\,\cdot\,}_2$ respectively.
$\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ are '''equivalent''' {{iff}} $\forall x \in R: \norm x_1 < 1 \iff \norm x_2 < 1$
\end{definition}