{
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  "body": "\n## Summary\n\nIn the mathematical field of set theory, an **ultrafilter** on a [[set|0BSJ7RQ9BW]] $X$ is a *maximal filter* on the set $X.$ In other words, it is a collection of subsets of $X$ that satisfies the definition of a filter on $X$ and that is maximal with respect to inclusion, in the sense that there does not exist a strictly larger collection of subsets of $X$ that is also a filter. (In the above, by definition a filter on a set does not contain the empty set.) Equivalently, an ultrafilter on the set $X$ can also be characterized as a filter on $X$ with the property that for every [[subset|0BJR9QR08E]] $A$ of $X$ either $A$ or its complement $X\\setminus A$ belongs to the ultrafilter.\n",
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