Cofinite filter

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

A subset $E$ of a set $X$ is cofinite in $X$ when its complement $X\setminus E$ is a finite set. For infinite $X$, the cofinite subsets form the cofinite filter. A nonprincipal ultrafilter on $\mathbb{N}$ contains every cofinite subset: it contains no singleton, hence no finite set, and decides between each set and its complement.