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  "body": "\n## Summary\n\nA filter on a set $X$ is a nonempty family of subsets of $X$ that does not contain $\\emptyset$.\nIt contains every superset of its members and the intersection of any two of its members.\nOne thinks of the members of a filter as the \"large\" sets, or as the sets that contain \"almost all\" points in some sense.\nFilters give a theory of [[convergence|70M1VAPFN0]] that works in every [[topological space|0CQHFPGJ45]], where sequences are not enough.\nMaximal filters are the [[ultrafilters|J64Y3JC0RE]].\n\n## Explanations\n\n### L1 human\n\nA filter is a way to say which sets are \"big enough\".\nFor a point $x$ of a topological space, the sets that contain a [[neighbourhood|0ZCY0Z80B6]] of $x$ form a filter: they are the sets that contain everything close enough to $x$.\nOn the natural numbers, the sets whose complement is finite form a filter: they contain all numbers from some point on.\nTwo big sets always overlap in a big set, and a set larger than a big set is big too.\n",
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