Principal filter

A filter on a set $A$ consisting of all subsets of $A$ containing a given subset $X$. If $X$ is a singleton $\{x\}$ then the principal filter on $\{x\}$ is a principal ultrafilter. The Fréchet filter is an example of a non-principal filter.

References

[1]	Thomas Jech, Set Theory (3rd edition), Springer (2003) ISBN 3-540-44085-2 Zbl 1007.03002