Complex sentence: 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.
Split into simpler sentences: The finite intersection property is useful in formulating an alternative definition of compactness. This formulation of compactness is used in some proofs of Tychonoff's theorem and the uncountability of the real numbers.