# Finite Intersection Property

A collection of subsets of a non-empty set is said to have the finite intersection property if every finite sub-collection of has a non-empty intersection.

In other words, the collection of subsets of the topological space is said to have a finite intersection property if every finite sub-collection of has a non-empty intersection, i.e. for any finite subset of , .

**Theorem**

A topological space is compact if and only if every collection of closed sets of , which satisfies the finite intersection property itself, has a non-empty intersection.