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.

$\blacksquare$


Also see