\begin{definition}[Definition:Equivalent Division Ring Norms/Topologically 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}} $d_1$ and $d_2$ are topologically equivalent metrics.
That is:
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, $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ are '''equivalent''' {{iff}} $\tau_1 = \tau_2$.
\end{definition}