In general topology, a branch of mathematics, a family ${\displaystyle A}$ 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.

Let ${\displaystyle X}$ be a set and ${\displaystyle {\mathcal {A}}}$ a family of subsets of ${\displaystyle X}$ (a subset of the power set of ${\displaystyle X}$). Then ${\displaystyle {\mathcal {A}}}$ is said to have the finite intersection property if the intersection of a finite number of subsets from ${\displaystyle {\mathcal {A}}}$ is always non-empty; it is said to have the strong finite intersection property if that intersection is always infinite.

In the study of filters, the intersection of a family of sets is called its 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 family with the finite intersection property.

If ${\displaystyle {\mathcal {A}}}$ has a non-empty kernel, then it has the finite intersection property trivially. The converse is false in general (although it holds trivially when ${\displaystyle {\mathcal {A}}}$ is finite). For example, the family of all cofinite subsets of a fixed infinite set — the Fréchet filter — has the finite intersection property, although its kernel is empty. More generally, any proper filter has the finite intersection property.

The finite intersection property is strictly stronger than requiring pairwise intersection to be non-empty, e.g., the family ${\displaystyle \{\{1,2\},\{2,3\},\{1,3\}\}}$ has non-empty pairwise intersections, but does not possess the finite intersection property. More generally, let ${\displaystyle n}$ be a natural number, let ${\displaystyle X}$ be a set with ${\displaystyle n}$ elements and let ${\displaystyle {\mathcal {A}}}$ consists of those subsets of ${\displaystyle X}$ which contain all elements but one. Then the intersection of fewer than ${\displaystyle n}$ subsets from ${\displaystyle {\mathcal {A}}}$ has non-empty intersection, but ${\displaystyle {\mathcal {A}}}$ lacks the finite intersection property.

If ${\displaystyle A_{0}\supseteq A_{1}\supseteq A_{2}\cdots }$ is a decreasing sequence of non-empty sets, then the family ${\textstyle {\mathcal {A}}=\left\{A_{0},A_{1},A_{2},\ldots \right\}}$ has the finite intersection property (and is even a π–system). If each ${\displaystyle A_{i}}$ is infinite, then ${\textstyle {\mathcal {A}}}$ admits the strong finite intersection property as well.