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.

$\blacksquare$

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