Let: :$m: \left({G, \tau}\right) \times \left({G, \tau}\right) \to \left({G, \tau}\right)$ be the mapping defined as: :$\forall \left({x, y}\right) \in G \times G: m \left({x, y}\right) = x \cdot y$ From the definition of topological group, $m$ is a continuous mapping. Let $\phi_1: \left({G, \tau}\right) \to \left({G, \tau}\right) \times \left({G, \tau}\right)$ be the mapping defined as: :$\phi_1 \left({x}\right) = \left({g, x}\right)$ Let $\phi_2: \left({G, \tau}\right) \to \left({G, \tau}\right) \times \left({G, \tau}\right)$ be the mapping defined as: :$\phi_2 \left({x}\right) = \left({x, g}\right)$ Let $\operatorname{pr}_1: \left({G, \tau}\right) \times \left({G, \tau}\right) \to \left({G, \tau}\right)$ be the first projection on $\left({G, \tau}\right) \times \left({G, \tau}\right)$: :$\forall \left({x, g}\right) \in G \times G: \operatorname{pr}_1 \left({x, g}\right) = x$ Let $\operatorname{pr}_2: \left({G, \tau}\right) \times \left({G, \tau}\right) \to \left({G, \tau}\right)$ be the second projection on $\left({G, \tau}\right) \times \left({G, \tau}\right)$: :$\forall \left({g, x}\right) \in G \times G: \operatorname{pr}_2 \left({g, x}\right) = x$ By inspection, it is seen that: :$\phi_1$ is the injective restriction to $g \times G$ of the inverse of $\operatorname{pr}_2$. :$\phi_2$ is the injective restriction to $G \times g$ of the inverse of $\operatorname{pr}_1$. Let $\phi_1 \left({x}\right) = \phi_1 \left({y}\right)$ for some $x, y \in G$. Then by definition of $\phi_1$: :$\left({g, x}\right) = \left({g, y}\right)$ By Equality of Ordered Pairs: :$x = y$ and so $\phi_1$ is injective. Similarly iet $\phi_2 \left({x}\right) = \phi_2 \left({y}\right)$ for some $x, y \in G$. Then by definition of $\phi_2$: :$\left({x, g}\right) = \left({y, g}\right)$ By Equality of Ordered Pairs: :$x = y$ and so $\phi_2$ is injective. Let $\left({g, y}\right) \in G \times G$. Then: :$\exists y \in G: \phi_1 \left({y}\right) = \left({g, y}\right)$ and so $\phi_1$ is surjective. Similarly let $\left({y, g}\right) \in G \times G$. Then: :$\exists y \in G: \phi_2 \left({y}\right) = \left({y, g}\right)$ and so $\phi_2$ is surjective. Thus as $\phi_1$ and $\phi_2$ are both injective and surjective, they are by definition both bijections. From Projection from Product Topology is Open, both $\operatorname{pr}_1$ and $\operatorname{pr}_2$ are open. From Bijection is Open iff Inverse is Continuous: :$\phi_1$ and $\phi_2$ are continuous. From Composite of Continuous Mappings is Continuous: :$m \circ \phi_1: \left({G, \tau}\right) \to \left({G, \tau}\right)$ and: :$m \circ \phi_2: \left({G, \tau}\right) \to \left({G, \tau}\right)$ are continuous mappings. By definition of left regular representation: :$\forall x \in G: L_g \left({x}\right) = g \cdot x = m \left({g, x}\right) = m \circ \phi_1 \left({x}\right)$ Similarly, by definition of right regular representation: :$\forall x \in G: R_g \left({x}\right) = x \cdot g = m \left({x, g}\right) = m \circ \phi_2 \left({x}\right)$ Thus $L_g$ and $R_g$ are continuous. By definition of inverse element of the group $G$: :$L_{g^{-1} } \left({x}\right) = g^{-1} \cdot x$ :$R_{g^{-1} } \left({x}\right) = x \cdot g^{-1}$ Thus for all $x \in G$: :$L_g \circ L_{g^{-1} } \left({x}\right) = g \cdot \left({g^{-1} \cdot x}\right) = x$ :$L_{g^{-1} } \circ L_g \left({x}\right) = g^{-1} \cdot \left({g \cdot x}\right) = x$ and similarly: :$R_g \circ R_{g^{-1} } \left({x}\right) = \left({x \cdot g^{-1} }\right) \cdot g = x$ :$R_{g^{-1} } \circ R_g \left({x}\right) = \left({x \cdot g}\right) \cdot g^{-1} = x$ So: :$L_g \circ L_{g^{-1} } = L_{g^{-1} } \circ L_g = I_G$ :$R_g \circ R_{g^{-1} } = R_{g^{-1} } \circ R_g = I_G$ where $I_G$ is the identity mapping on $G$. Thus by definition, $L_{g^{-1} }$ and $R_{g^{-1} }$ are the inverses of $L_g$ and $R_g$ respectively. As $L_{g^{-1} }$ and $R_{g^{-1} }$ are themselves regular representations with respect to elements of $G$, they are themselves continuous bijections. So $L_g$ and $R_g$ are continuous bijections whose inverses are likewise continuous bijections. Hence, by definition, $L_g$ and $R_g$ are homeomorphisms. {{qed}} Category:Homeomorphisms Category:Regular Representations Category:Topological Groups 8uw0cp2tyyx5oa1q6pm3eroocis6btm