Finite intersection property

In general topology, a branch of mathematics, a collection A of subsets of a set X is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of A is nonempty. It has the strong finite intersection property (SFIP) if the intersection over any finite subcollection of A is infinite.

A centered system of sets is a collection of sets with the finite intersection property.

Let ${\displaystyle X}$ be a set with ${\displaystyle A=\{A_{i}\}_{i\in I}}$ a family of subsets of ${\displaystyle X}$. Then the collection ${\displaystyle A}$ has the finite intersection property (FIP), if any finite subcollection ${\displaystyle J\subseteq I}$ has non-empty intersection ${\displaystyle \bigcap _{i\in J}A_{i}.}$

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