Multiplicative Regular Representations of Units of Topological Ring are Homeomorphisms

Theorem
Let $\struct{R, +, \circ, \tau}$ be a topological ring with unity $1_R$.

Let $x \in R$ be a unit of $R$.

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

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