Additive Regular Representations of Topological Ring are Homeomorphisms

Theorem
Let $\struct{R, +, \circ, \tau}$ be a topological ring.

Let $x \in R$.

Let $\lambda^+_x$ and $\rho^+_x$ be the left and right regular representations of $\struct{R, +}$ with respect to $x$.

Then $\,\lambda^+_x, \,\rho^+_x : \struct{R, \tau} \to \struct{R, \tau}$ are homeomorphisms.