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