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  "body": "\n## Definition\n\nLet $X$ be a set.\nAn *ultrafilter* on $X$ is a [[filter|15PF73BKY6]] $\\mathcal{U}$ on $X$ such that for every filter $\\mathcal{F}$ on $X$ with $\\mathcal{U} \\subseteq \\mathcal{F}$, one has $\\mathcal{F} = \\mathcal{U}$.\n\n## Remarks\n\nA filter $\\mathcal{U}$ on $X$ is an ultrafilter if and only if, for every $A \\subseteq X$, either $A \\in \\mathcal{U}$ or $X \\setminus A \\in \\mathcal{U}$.\n\nExample: for $a \\in X$, the principal filter $\\mathcal{U}_a = \\set{A \\subseteq X : a \\in A}$ is an ultrafilter.\nFor each $A \\subseteq X$, either $a \\in A$ or $a \\in X \\setminus A$.\n\nNon-example: on $\\N$, the Fréchet filter of sets with finite complement is not an ultrafilter.\nThe set $E$ of even numbers and its complement are both infinite, so neither has finite complement.\n\nBy [[Zorn's lemma|04RXTQA5ZA]], every filter on $X$ is contained in an ultrafilter.\nOn an infinite set, an ultrafilter that contains the Fréchet filter is not principal, so non-principal ultrafilters exist.\n",
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  "name": "Ultrafilter",
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