Definition:Topology Induced by Division Ring Norm

Definition
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring whose norm is $\norm {\,\cdot\,}$.

The topology on $\struct {R, \norm {\,\cdot\,} }$ induced by $\norm {\,\cdot\,}$ is defined as the topology $\tau$ generated by the basis consisting of the set of all open balls of $\struct {R, \norm {\,\cdot\,} }$.

Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$.

By the definition of the metric induced by a norm on division ring it follows that the topology on the normed division ring $\struct{R, \norm{\,\cdot\,}}$ induced by (the norm) $\norm{\,\cdot\,}$ is the topology induced by the metric $d$.

Also known as
The topological space which is so induced is also known as the topological space associated with the (given) norm.

When the context is clear, the phrase norm topology for $\norm{\,\cdot\,}$ can be used.

The normed division ring whose norm induces this topology can be said to give rise to the topological space.

Also see

 * Metric Induced by Norm on Normed Division Ring is Metric in which it is shown that $d$ is, in fact, a metric on $R$
 * Metric Induces Topology, in which it is shown that $\tau$ is, in fact, a topology on $\struct {R, d}$