Category:Topological Division Rings

This category contains results about Topological Division Rings.

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 $R \setminus \set {0_R}$.