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.
$\blacksquare$