Federer–Morse theorem explained

Federer–Morse theorem should not be confused with Morse–Sard–Federer theorem.

In mathematics, the Federer–Morse theorem, introduced by, 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

References

• • Book: Fabec, Raymond C.. Fundamentals of Infinite Dimensional Representation Theory. 2000. CRC Press. 978-1-58488-212-1. registration.
  • Book: Parthasarathy, K. R.. Probability measures on metric spaces. Probability and Mathematical Statistics. 3. Academic Press, Inc.. New York-London. 1967.

Further reading

• L. W. Baggett and Arlan Ramsay, A Functional Analytic Proof of a Selection Lemma, Can. J. Math., vol. XXXII, no 2, 1980, pp. 441–448.

Notes and References

1. Section 4 of .
2. Page 12 of