Federer–Morse theorem

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Federer–Morse theorem (1943) says: If f is an onto continuous map from a compact metric space X to another compact metric space Y, then there exists a Borel subset Z of X such that f restricted to Z is a bijection from Z onto Y. The inverse map, from Y back to Z, is also Borel measurable; in other words, it is a Borel section of f and makes f|_Z a Borel isomorphism between Z and Y. Intuitively, you can pick a nicely measurable slice Z of X that covers Y exactly once under f.