Finite intersection property

In general topology, a branch of mathematics, a family ${\displaystyle {\mathcal {A}}}$ of subsets of a set ${\displaystyle X}$ is said to have the finite intersection property (FIP) if any finite subfamily of ${\displaystyle {\mathcal {A}}}$ has non-empty intersection. It has the strong finite intersection property (SFIP) if any finite subfamily has infinite intersection. 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.