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