Definition:Equivalent Division Ring Norms/Topologically 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 $d_1$ and $d_2$ are topologically equivalent metrics

Let $\tau_1$ and $\tau_2$ be the topologies induced by the norms $\norm{\,\cdot\,}_1$ and $\norm{\,\cdot\,}_2$ respectively.

By the definition of a topology induced by a norm then $\norm{\,\cdot\,}_1$ and $\norm{\,\cdot\,}_2$ are equivalent $\tau_1 = \tau_2$