Definition:Principal Ultrafilter

Definition
Let $S$ be a set.

Let $\mathcal P \left({S}\right)$ denote the power set of $S$.

Let $\mathcal F \subset \mathcal P \left({S}\right)$ be an ultrafilter on $S$ with a cluster point.

Then $\mathcal F$ is a principal ultrafilter on $S$.

Also known as
A principal ultrafilter is also known as a fixed ultrafilter.

Also see

 * From Cluster Point of Ultrafilter is Unique, an ultrafilter is either principal or nonprincipal -- there are no other options.


 * In Principal Ultrafilter is All Sets Containing Cluster Point it is shown that a principal ultrafilter on $S$ whose cluster point is $x$ is the set of all subsets of $S$ which contain $x$.