A question on the right translation

Question

Here is a claim:

Let $G$ be a right topological group and $g$ be any element of $G$. Then the right translation $R_g$ of $G$ by $g$ is a homeomorphism of the space $G$ onto itself.

How can I reach this conclusion?

Answer 1

By definition of a right topological group the right translation $R_g$ is assumed to be continuous.

Moreover the inverse of $R_g$ is just $R_{g^{-1}}$, which is another right translation and hence also continuous.

Therefore $R_g$ is an homeomorphism.