Example 1

Let $K$ be an arbitrary division ring. For each $x \in K$, put $|x| = 1$ if $x \neq 0$, and $|0| = 0$. The mapping $x \to |x|$ so defined is an absolute value on $K$, called the improper absolute value. The topology defined by an absolute value $|x|$ on a division ring $K$ is discrete if and only if $|x|$ is the improper absolute value. This condition is clearly sufficient; conversely, if the topology of $K$ is discrete, $|x|$ can take no value $\alpha > 0$ other than 1; for if we had $|x_0| = \alpha < 1$, the sequence $(x_0^n)$ would consist of non-zero terms and would converge to 0; to deal with the case $\alpha > 1$, consider $x_0^{-1}$ in place of $x_0$.