Uniform structure of a topological group

Remarks

Both uniform structures induce the given topology of $G$: the neighbourhoods of $x$ are the sets $xU$ for the left one and $Ux$ for the right one, with $U \in \mathcal{N}$. Example: for $(\R, +)$, the set $L_U$ with $U = (-\varepsilon, \varepsilon)$ is $\set{(x, y) : \abs{y - x} < \varepsilon}$, so both uniformities are the uniformity of the usual metric. If $G$ is abelian, then $L_U = R_U$ for every $U$, so the two structures coincide. In general they differ; the inversion map $x \mapsto x^{-1}$ carries one to the other. Some authors also use the two-sided uniformity, generated by the sets $L_U \cap R_U$.