# Category:Topological Division Rings

Jump to navigation
Jump to search

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

## Pages in category "Topological Division Rings"

The following 7 pages are in this category, out of 7 total.

### N

- Normed Division Ring Operations are Continuous
- Normed Division Ring Operations are Continuous/Addition
- Normed Division Ring Operations are Continuous/Corollary
- Normed Division Ring Operations are Continuous/Inversion
- Normed Division Ring Operations are Continuous/Multiplication
- Normed Division Ring Operations are Continuous/Negation