A family of sets has the finite intersection property when every finite subfamily has nonempty intersection. A topological space is compact exactly when every family of closed subsets with the finite intersection property has nonempty total intersection.
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The Finite Intersection Property (FIP) is a concept from topology and set theory. It applies to a collection of sets and states that a family of sets has the finite intersection property if the intersection of every finite subcollection of these sets is non-empty. Formally, let \( \mathcal{A} \) be a collection of sets.