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  "body": "\n## Definition\n\nLet $(X, \\mathcal{U})$ be a [[uniform space|A54T86KS63]].\nAn *entourage* of $X$ is a member of the uniformity $\\mathcal{U}$, that is, a subset $U \\subseteq X \\times X$ with $U \\in \\mathcal{U}$.\nFor an entourage $U$ and a point $x \\in X$, the set $U[x] = \\set{y \\in X : (x, y) \\in U}$ is the $U$-neighbourhood of $x$.\nAn entourage $U$ is *symmetric* if $(x, y) \\in U$ implies $(y, x) \\in U$.\n\n## Remarks\n\nOn $\\R$ with the uniformity of the metric $\\abs{x - y}$, the set $\\set{(x, y) : \\abs{x - y} \u003c 1}$ is an entourage, and $U[x] = (x - 1, x + 1)$.\nThe set $\\set{(x, y) : \\abs{x - y} \u003c 1 / (1 + x^2)}$ contains the diagonal, but it is not an entourage of that uniformity.\nIt contains no set $\\set{(x, y) : \\abs{x - y} \u003c \\varepsilon}$ with $\\varepsilon \u003e 0$, because its width tends to $0$ as $\\abs{x}$ grows.\nEvery entourage contains a symmetric entourage, namely $U \\cap U^{-1}$.\nThe sets $U[x]$, for $U$ an entourage, form a neighbourhood base at $x$ of the uniform topology.\n\n## Explanations\n\n### L1 human\n\nAn entourage is a set of pairs of points that a uniform space counts as close.\nPicture the plane $X \\times X$; an entourage is a region that contains the whole diagonal $y = x$.\nFor $\\R$ with its usual distance, the region between the lines $y = x - 1$ and $y = x + 1$ is an entourage.\n",
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