Definition:Topological Division Ring

Definition

Let $\left({R, +, \circ}\right)$ be a division ring with zero $0_R$.

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

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

$\phi \left({x}\right) = x^{-1}$ for each $x \in R \setminus \left\{{0_R}\right\}$

Then $\left({R,+,\circ,\tau}\right)$ is a topological division ring if and only if:

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

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