Combination Theorem for Continuous Mappings/Topological Division Ring/Product Rule
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Theorem
Let $\struct {S, \tau_{_S} }$ be a topological space.
Let $\struct {R, +, *, \tau_{_R} }$ be a topological division ring.
Let $f, g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ be continuous mappings.
Let $f * g: S \to R$ be the mapping defined by:
- $\forall x \in S: \map {\paren {f * g} } x = \map f x * \map g x$
Then:
- $f * g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.
Proof
By definition of a topological division ring:
- $\struct {R, +, *, \tau_{_R} }$ is a topological ring.
From Product Rule for Continuous Mappings to Topological Ring:
- $f * g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a continuous mapping.
$\blacksquare$