# Finite Intersection Property

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

In other words, the collection $$A = \left\{ {{A_\alpha }:\alpha \in I} \right\}$$ of subsets of the topological space $$X$$ is said to have a finite intersection property if every finite sub-collection of $$A$$ has a non-empty intersection, i.e. for any finite subset $${I_1}$$ of $$I$$, $$\bigcap\limits_{\beta \in {I_1}} {{A_\beta }} \ne \phi $$.

**Theorem**

A topological space $$X$$ is compact if and only if every collection $$A = \left\{ {{A_\alpha }:\alpha \in I} \right\}$$ of closed sets of $$X$$, which satisfies the finite intersection property itself, has a non-empty intersection.