Ultrafilter on a set

Summary

In the mathematical field of set theory, an ultrafilter on a set $X$ 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 $A$ of $X$ either $A$ or its complement $X\setminus A$ belongs to the ultrafilter.