A {{pi}}-system is a family of sets that is closed under finite intersections of one or more of its sets. For a family of sets , the family of sets which is all finite intersections of one or more sets from , is called the {{pi}}-system generated by $\mathcal{A/math>,}} because it is the smallest -system having as a subset.$

The upward closure of in is the set For any family $\mathcal{A/math>,}} the finite intersection property is equivalent to any of the following:$