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  "body": "\n## Definition\n\nLet $G$ be a [[topological group|92700QAB5Y]] with identity $e$, and let $\\mathcal{N}$ be the set of [[neighbourhoods|0ZCY0Z80B6]] of $e$.\nFor $U \\in \\mathcal{N}$ put\n$$L_U = \\set{(x, y) \\in G \\times G : x^{-1} y \\in U}, \\qquad R_U = \\set{(x, y) \\in G \\times G : y x^{-1} \\in U}.$$\nThe *left uniform structure* of $G$ is the [[uniform structure|A54T86KS63]] on $G$ whose [[entourages|FVFFKNKEDF]] are the subsets of $G \\times G$ that contain some $L_U$ with $U \\in \\mathcal{N}$.\nThe *right uniform structure* of $G$ is defined in the same way with the sets $R_U$.\n\n## Remarks\n\nBoth 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 $U \\in \\mathcal{N}$.\nExample: for $(\\R, +)$, the set $L_U$ with $U = (-\\varepsilon, \\varepsilon)$ is $\\set{(x, y) : \\abs{y - x} \u003c \\varepsilon}$, so both uniformities are the uniformity of the usual metric.\nIf $G$ is abelian, then $L_U = R_U$ for every $U$, so the two structures coincide.\nIn general they differ; the inversion map $x \\mapsto x^{-1}$ carries one to the other.\nSome authors also use the two-sided uniformity, generated by the sets $L_U \\cap R_U$.\n",
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