# 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 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 finite intersection property if every finite sub-collection of $A$ has 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.