Finite intersection property

In general topology, a branch of mathematics, a non-empty family A of subsets of a set is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of is non-empty. It has the strong finite intersection property (SFIP) if the intersection over any finite subcollection of is infinite. Sets with the finite intersection property are also called centered systems and filter subbases. The finite intersection property can be used to reformulate topological compactness in terms of closed sets; this is its most prominent application. Other applications include proving that certain perfect sets are uncountable, and the construction of ultrafilters. Let be a set and a nonempty family of subsets of ; that is, is a subset of the power set of . Then is said to have the finite intersection property if every nonempty finite subfamily has nonempty intersection; it is said to have the strong finite intersection property if that intersection is always infinite. In symbols, has the FIP if, for any choice of a finite nonempty subset of , there must exist a point Likewise, has the SFIP if, for every choice of such , there are infinitely many such . In the study of filters, the common intersection of a family of sets is called a kernel, from much the same etymology as the sunflower. Families with empty kernel are called free; those with nonempty kernel, fixed. The empty set cannot belong to any collection with the finite intersection property. A sufficient condition for the FIP intersection property is a nonempty kernel. The converse is generally false, but holds for finite families; that is, if is finite, then has the finite intersection property if and only if it is fixed. The finite intersection property is strictly stronger than pairwise intersection; the family has pairwise intersections, but not the FIP. More generally, let be a positive integer greater than unity, , and . Then any subset of with fewer than elements has nonempty intersection, but lacks the FIP.