Definition:Topological Division Ring

Definition

Let $\struct {R, +, \circ}$ be a division ring with zero $0_R$.

Let $\tau$ be a topology on $R$.

Let the mapping $\phi: R \setminus \set {0_R} \to R$ be defined as:

$\map \phi x = x^{-1}$ for each $x \in R \setminus \set {0_R}$

Then $\struct {R, +, \circ, \tau}$ is a topological division ring if and only if:

$(1): \quad \struct {R, +, \circ, \tau}$ is a topological ring

$(2): \quad \phi$ is a $\tau'$-$\tau$-continuous mapping, where $\tau'$ is the $\tau$-relative subspace topology on $R \setminus \set {0_R}$.

Also see

• Definition:Topological Field

• Results about topological division rings can be found here.

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