Combination Theorem for Continuous Mappings/Topological Semigroup
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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 $f,g : \struct{S, \tau_{_S}} \to \struct{G, \tau_{_G}}$ be continuous mappings.
Then the following results hold:
Product Rule
- $f * g: \struct{S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping.
Multiple Rule
- $\lambda * f: \struct {S, \tau_S} \to \struct {G, \tau_G}$ is a continuous mapping
- $f * \lambda: \struct {S, \tau_S} \to \struct {G, \tau_G}$ is a continuous mapping.