Combination Theorem for Continuous Mappings/Normed Division Ring
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Theorem
Let $\struct{S, \tau_{_S}}$ be a topological space.
Let $\struct{R, +, *, \norm{\,\cdot\,}}$ be a normed division ring.
Let $\tau_{_R}$ be the topology induced by the norm $\norm{\,\cdot\,}$.
Let $\lambda \in R$.
Let $f,g : \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ be continuous mappings.
Let $U = S \setminus \set{x : \map g x = 0}$
Let $g^{-1} : U \to R$ denote the mapping defined by:
- $\forall x \in U : \map {g^{-1}} x = \map g x^{-1}$
Let $\tau_{_U}$ be the subspace topology on $U$.
Then the following results hold:
Sum Rule
- $f + g: \struct {S, \tau_{_S} } \to \struct{R, \tau_{_R} }$ is continuous.
Translation Rule
- $\lambda + f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.
Negation Rule
- $- g : \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ is continuous.
Product Rule
- $f * g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.
Multiple Rule
- $\lambda * f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous
- $f * \lambda: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.
Inverse Rule
- $g^{-1}: \struct {U, \tau_{_U} } \to \struct {R, \tau_{_R} }$ is continuous.