Combination Theorem for Continuous Mappings/Normed Division Ring/Translation Rule
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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 $f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ be a continuous mapping.
Let $\lambda + f : S \to R$ be the mapping defined by:
- $\forall x \in S: \map {\paren {\lambda + f} } x = \lambda + \map f x$
Then
- $\lambda + f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.
Proof
From Corollary to Normed Division Ring Operations are Continuous:
- $\struct{R, +, *, \tau_{_R} }$ is a topological division ring.
From Translation Rule for Continuous Mappings to Topological Division Ring:
- $\lambda + f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a continuous mapping.
$\blacksquare$