Finite intersection property

In general topology, a branch of mathematics, a non-empty family A of subsets of a set ${\displaystyle X}$ is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of ${\displaystyle A}$ is non-empty. It has the strong finite intersection property (SFIP) if the intersection over any finite subcollection of ${\displaystyle A}$ is infinite. Sets with the finite intersection property are also called centered systems and filter subbases.^[1]

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.