Definition 3

A valued division ring is a division ring $K$ endowed with the structure defined by a given absolute value on $K$.

A valued division ring will always be considered as endowed with the topology defined by its absolute value, which makes it a topological division ring. If $K_0$ is a division subring of a valued division ring $K$, the restriction to $K_0$ of the absolute value on $K$ is an absolute value on $K_0$, which defines on $K_0$ the topology induced by the topology of $K$.