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