Combination Theorem for Continuous Mappings/Topological Group/Product Rule

Theorem

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

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

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

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

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

Then:

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

Proof

By definition, a topological group is a topological semigroup.

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

From Product Rule for Continuous Mappings to Topological Semigroup, $f * g : \struct{S, \tau_{_S}} \to \struct{G, \tau_{_G}}$ is a continuous mapping.

$\blacksquare$