Category:Topological Division Rings
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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.
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- 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