# 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 $\struct {R, d}$

## Sources

- 1997: Fernando Q. Gouvea:
*p-adic Numbers: An Introduction*: $\S 2.3$ Topology, Definition $2.3.1$