Combination Theorem for Continuous Mappings/Topological Group
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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 $f, g : \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ 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.
Inverse Rule
- $g^{-1}: \struct {S, \tau_S} \to \struct {G, \tau_G}$ is a continuous mapping.