Combination Theorem for Continuous Mappings/Topological Semigroup

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.