The first two properties imply that a filter on a set has the finite intersection property.

A filter has the finite intersection property by definition.

The proof is similar to the preceding statement; the finite intersection property takes the role played by completeness.

A topological proof for the uncountability of the real numbers is described at finite intersection property.

Let , , F having the finite intersection property.

A centered system of sets is a collection of sets with the finite intersection property.

Any collection of closed subsets of X with the finite intersection property has nonempty intersection.

A family of sets A has the strong finite intersection property (sfip), if every finite subfamily of A has infinite intersection.

In general topology, a branch of mathematics, a collection A of subsets of a set X is said to have the finite intersection property if the intersection over any finite subcollection of A is nonempty.

Likewise, it is analogous to the finite intersection property characterization of compactness in topological spaces: a collection of closed sets in a compact space has a non-empty intersection if every finite subcollection has a non-empty intersection.

The finite intersection property is useful in formulating an alternative definition of compactness: a space is compact if and only if every collection of closed sets satisfying the finite intersection property has nonempty intersection itself.

One can show that every filter of a Boolean algebra (or more generally, any subset with the finite intersection property) is contained in an ultrafilter (see Ultrafilter lemma) and that free ultrafilters therefore exist, but the proofs involve the axiom of choice in the form of Zorn's Lemma.