Federer–Morse theorem

Not to be confused with Morse–Sard–Federer theorem.

In mathematics, the Federer–Morse theorem, introduced by Federer and Morse (1943), states that if f is a surjective continuous map from a compact metric space X to a compact metric space Y, then there is a Borel subset Z of X such that f restricted to Z is a bijection from Z to Y.^[1] Moreover, the inverse of that restriction is a Borel section of f—it is a Borel isomorphism.^[2]

See also

• Uniformization
• Hahn–Banach theorem