Filter on a set

A filter $\mathcal{F}$ on a set $X$ is a nonempty family of subsets that excludes the empty set, is closed under finite intersections, and is upward closed under inclusion.

The cofinite filter on an infinite set consists of all subsets whose complements are finite.

An ultrafilter on a set $X$ is a proper filter on a set that contains exactly one of $A$ and $X\setminus A$ for every $A\subseteq X$.

The principal ultrafilter at $x\in X$ is $\{A\subseteq X:x\in A\}$. An ultraproduct by a principal ultrafilter is isomorphic to the factor indexed by $x$.

A nonprincipal ultrafilter contains no finite set. On an infinite set it contains the cofinite filter.

The Stone-Cech compactification $\beta \mathbb{N}$ can be identified with the space of ultrafilters on $\mathbb{N}$. Its basic clopen sets are $\bar{A}=\{\mathcal{U}:A\in \mathcal{U}\}$.