Combination Theorem for Continuous Mappings/Topological Group/Inverse Rule

Theorem
Let $\struct{S, \tau_S}$ be a topological space.

Let $\struct{G, *, \tau_G}$ be a topological group.

Let $g : \struct{S, \tau_S} \to \struct{G, \tau_G}$ be a continuous mapping.

Let $g^{-1} : S \to G$ be the mapping defined by:
 * $\forall x \in S: \map {\paren{g^{-1}}} x = \map g x^{-1}$

Then:
 * $g^{-1} : \struct{S, \tau_S} \to \struct{G, \tau_G}$ is a continuous mapping.