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.