Definition:Principal Ultrafilter

Definition
Let $$X$$ be a set, and $$\mathcal P \left({X}\right)$$ be the power set of $$X$$.

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

Then $$\mathcal F$$ is a principal ultrafilter or a fixed ultrafilter on $$X$$.

Nonprincipal Ultrafilter
If $$\mathcal F \subset \mathcal P \left({X}\right)$$ is an ultrafilter on $$X$$ which does not have a cluster point, it is a nonprincipal ultrafilter or a free ultrafilter on $$X$$.

Also see

 * From Ultrafilter has One Cluster Point at Most, 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 whose cluster point is $$x$$ is the set of all subsets of $$X$$ which contain $$x$$.