Exercise 3, § 3

Show that the following conditions on a topological group $G$ are equivalent: $\alpha)$ The left and right uniformities on $G$ are the same. $\beta)$ If $V$ is any neighbourhood of $e$ in $G$, there is a neighbourhood $W$ of $e$ such that $xWx^{-1} \subset V$ for all $x \in G$. $\gamma)$ The symmetry $x \to x^{-1}$ is a uniformly continuous mapping of $G_d$ onto $G_d$ (or of $G_s$ onto $G_s$). $\delta)$ The mapping $(x, y) \to xy$ is a uniformly continuous mapping of one of the four spaces $G_d \times G_d, G_s \times G_s, G_d \times G_s, G_s \times G_d$ into one of the two spaces $G_d, G_s$.