Ultrafilter

Summary

An ultrafilter on a set $X$ is a filter on $X$ that is maximal: no strictly larger filter on $X$ exists. Equivalently, for every subset $A$ of $X$, either $A$ or its complement belongs to it. Every filter lies in an ultrafilter, by Zorn's lemma. Ultrafilters describe compactness, build the Stone–Čech compactification, and give ultraproducts in logic.