Combination Theorem for Continuous Mappings/Topological Semigroup/Multiple Rule

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Theorem

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

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


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

Let $c_\lambda : S \to G$ be the constant mapping defined by:

$\forall x \in S: \map {c_\lambda} x = \lambda$

From Constant Mapping is Continuous, $c_\lambda$ is continuous.

From Product Rule for Continuous Mappings to Topological Semigroup:

$c_\lambda * f$ and $f * c_\lambda$ are continuous.

Now:

\(\, \displaystyle \forall x \in S : \, \) \(\displaystyle \map {\paren {c_\lambda * f} } x\) \(=\) \(\displaystyle \map {c_\lambda} x * \map f x\) Definition of $c_\lambda * f$
\(\displaystyle \) \(=\) \(\displaystyle \lambda * \map f x\) Definition of $c_\lambda$
\(\displaystyle \) \(=\) \(\displaystyle \map {\paren {\lambda * f} } x\) Definition of $\lambda * f$

From Equality of Mappings:

$c_\lambda * f = \lambda * f$

Similarly:

$f * c_\lambda = f * \lambda$

The result follows.

$\blacksquare$