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  "body": "\n## Summary\n\nA Cauchy filter on a [[uniform space|A54T86KS63]] is a [[filter|15PF73BKY6]] that contains arbitrarily small sets, where \"small\" is measured by the [[entourages|FVFFKNKEDF]].\nIt is the uniform version of a [[Cauchy sequence|0PWX18FM4G]].\nA uniform space is complete when every Cauchy filter converges, and this gives completeness without a metric.\n\n## Explanations\n\n### L1 human\n\nA Cauchy sequence is a sequence whose terms eventually all lie very close together.\nA filter is a family of \"large\" sets.\nA Cauchy filter has, for each degree of closeness, a member whose points are all that close to each other.\nIn $\\R$, the filter of all sets that contain an interval $(-\\varepsilon, \\varepsilon)$ for some $\\varepsilon \u003e 0$ is a Cauchy filter.\nIn the rational numbers, the filter of neighbourhoods of $\\sqrt{2}$ traced on $\\Q$ is a Cauchy filter that does not converge in $\\Q$.\n",
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