The left and right uniformities of a topological group

Let $G$ be a topological group with identity $e$. For every neighbourhood $U$ of $e$ (Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open), put

$L_U=\{(x,y):x^{-1}y\in U\},\qquad R_U=\{(x,y):yx^{-1}\in U\}.$

The filters generated by the $L_U$ and by the $R_U$ are respectively the left uniformity and the right uniformity of $G$. These are uniformities. Indeed, finite intersections of identity neighbourhoods refine finite intersections of the corresponding relations; the diagonal lies in every relation; $L_U^{-1}=L_{U^{-1}}$ and $R_U^{-1}=R_{U^{-1}}$; and continuity of multiplication at $(e,e)$ gives a neighbourhood $V$ with $V\cdot V\subseteq U$, whence $L_V\circ L_V\subseteq L_U$ and $R_V\circ R_V\subseteq R_U$. Thus all axioms of Uniform space in the entourage formulation hold.