# Definition:Topology Induced by Division Ring Norm

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## 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 phase 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.