---
tag: SWP7PEZX30
kind: concept
name: Ultrafilter on a set
aliases: [countably-complete-ultrafilter, free-ultrafilter-set-theory, principal-ultrafilter, ramsey-ultrafilter, rudin-keisler-equivalent, rudin-keisler-order, rudin-keisler-ordering, selective-ultrafilter, ultra-prefilter, ultrafilter-set-theory, ultrafilter-monad, ultrafilter-principle]
sources:
  - wikipedia: en:67416519@r1361725976
    locator: Ultrafilter on a set
    note: adopted
    mention: wikipedia://en/67416519
    sections: [Summary]
same_as:
  wikidata: Q106671513
  wikipedia: Ultrafilter on a set
inventory: [XMXMY4SR1M]
---

## Summary

In 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.