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  "body": "\n## Definition\n\nLet $X$ be a set.\nA *filter* on $X$ is a family $\\mathcal{F}$ of subsets of $X$ such that:\n\n1. $X \\in \\mathcal{F}$.\n2. $\\emptyset \\notin \\mathcal{F}$.\n3. If $A \\in \\mathcal{F}$ and $A \\subseteq B \\subseteq X$, then $B \\in \\mathcal{F}$.\n4. If $A \\in \\mathcal{F}$ and $B \\in \\mathcal{F}$, then $A \\cap B \\in \\mathcal{F}$.\n\n## Remarks\n\nSome authors, and Mathlib, drop clause 2.\nThen the only family that satisfies clauses 1, 3 and 4 and contains $\\emptyset$ is the [[power set|00E0YZCERH]] of $X$.\nA filter in the sense above is then called a *proper* filter.\n\nExample: for a point $x$ of a [[topological space|0CQHFPGJ45]] $X$, the [[neighbourhoods|0ZCY0Z80B6]] of $x$ form a filter $\\mathcal{N}(x)$, the *neighbourhood filter* of $x$.\nClause 2 holds because each neighbourhood contains $x$, and clause 4 holds because the intersection of two neighbourhoods is a neighbourhood.\n\nExample: on an infinite set $X$, the sets with finite complement form a filter, the *Fréchet filter*.\nClause 2 holds because $X$ is infinite, and clause 4 holds because the union of two finite sets is finite.\n\nExample: for $a \\in X$, the family $\\set{A \\subseteq X : a \\in A}$ is a filter, the *principal filter* at $a$.\n\nNon-example: on $\\N$, the family of sets that contain $0$ or contain $1$ satisfies clauses 1, 2 and 3.\nIt fails clause 4: it contains $\\set{0}$ and $\\set{1}$, but not $\\set{0} \\cap \\set{1} = \\emptyset$.\n\nA nonempty family $\\mathcal{B}$ of nonempty subsets of $X$ is a *filter base* if the intersection of any two members contains a member.\nThe supersets of members of a filter base form a filter.\n",
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