Definition:Equivalent Division Ring Norms/Open Unit Ball Equivalent

Definition
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 $\forall x \in R: \norm{x}_1 \lt 1 \iff \norm{x}_2 \lt 1$