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Finite intersection property

In topology, the finite intersection property is a property of a collection of subsets of a set X. A collection has this property if the intersection over any finite subcollection of the collection is nonempty.

This is trivially satisfied if the intersection over the entire collection is nonempty, and is also trivially satisfied if the collection is nested, meaning a single element of any finite subcollection contains all the remaining ones, as in the sequence (0,1/n). These are not the only possibilities however. For example, if our set X is (0,1), and Xi are those elements of X with the digit 0 in the i'th decimal place, then any finite intersection is nonempty (just take 0 in those finitely many places and 1 in the rest), but the intersection of all Xi for i≥1 is empty, since no element of (0,1) has all zero digits.

The finite intersection property is useful in formulating an alternative definition of compactness, which in turn is used in some proofs of Tychonoff's theorem and the uncountability of the real numbers.