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  "body": "\n## Definition\n\nLet $(X, \\mathcal{U})$ be a [[uniform space|A54T86KS63]] and let $\\mathcal{F}$ be a [[filter|15PF73BKY6]] on $X$.\nThe filter $\\mathcal{F}$ is a *Cauchy filter* if for every [[entourage|FVFFKNKEDF]] $U \\in \\mathcal{U}$ there is a set $A \\in \\mathcal{F}$ with $A \\times A \\subseteq U$.\nThe uniform space is *complete* if every Cauchy filter on $X$ converges to a point of $X$.\n\n## Remarks\n\nThe neighbourhood filter $\\mathcal{N}_x$ of a point $x$ is a Cauchy filter.\nGiven an entourage $U$, choose a symmetric entourage $V$ with $V \\circ V \\subseteq U$ and let $A = V[x]$, a member of $\\mathcal{N}_x$.\nFor $y, z \\in A$ one has $(y, x) \\in V$ and $(x, z) \\in V$, so $(y, z) \\in V \\circ V \\subseteq U$.\nOn $\\R$ with its usual uniformity, the filter of sets that contain some interval $[n, \\infty)$ is not a Cauchy filter.\nFor $U = \\set{(x, y) : \\abs{x - y} \u003c 1}$, every member contains two points at distance $1$, so no member $A$ has $A \\times A \\subseteq U$.\nIn a metric space, a sequence is a [[Cauchy sequence|0PWX18FM4G]] if and only if the filter of its tails is a Cauchy filter.\nEvery convergent filter is a Cauchy filter.\n\n## Explanations\n\n### L1 human\n\nA Cauchy filter is a filter with members as small as one likes, in the sense of the uniform structure.\nIt is the right replacement for Cauchy sequences in spaces where sequences are not enough.\n",
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