Combination Theorem for Continuous Mappings/Topological Group/Multiple Rule

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Theorem

Let $\struct{S, \tau_{_S}}$ be a topological space.

Let $\struct{G, *, \tau_{_G}}$ be a topological group.


Let $\lambda \in G$.

Let $f : \struct{S, \tau_{_S}} \to \struct{G, \tau_{_G}}$ be a continuous mapping.


Let $\lambda * f : S \to G$ be the mapping defined by:

$\forall x \in S: \map {\paren{\lambda * f}} x = \lambda * \map f x$

Let $f * \lambda : S \to G$ be the mapping defined by:

$\forall x \in S: \map {\paren{f * \lambda}} x = \map f x * \lambda$


Then:

$\lambda * f : \struct{S, \tau_{_S}} \to \struct{G, \tau_{_G}}$ is a continuous mapping
$f * \lambda : \struct{S, \tau_{_S}} \to \struct{G, \tau_{_G}}$ is a continuous mapping.


Proof

By definition, a topological group is a topological semigroup.

Hence $\struct{G, *, \tau_{_G}}$ is a topological semigroup.

From Multiple Rule for Continuous Mappings to Topological Semigroup, $\lambda * f, f * \lambda : \struct{S, \tau_{_S}} \to \struct{G, \tau_{_G}}$ are continuous mappings.

$\blacksquare$

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