Federer–Morse_theorem

Federer–Morse theorem

Federer–Morse theorem

On a property of surjective continuous maps between compact metric spaces

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

References

  1. [1]
    Section 4 of Parthasarathy (1967).
  2. [2]
    Page 12 of Fabec (2000)
  • Baggett, Lawrence W. (1990), "A Functional Analytical Proof of a Borel Selection Theorem", Journal of Functional Analysis, 94: 437–450
  • Fabec, Raymond C. (2000). Fundamentals of Infinite Dimensional Representation Theory. CRC Press. ISBN 978-1-58488-212-1.
  • Federer, Herbert; Morse, A. P. (1943), "Some properties of measurable functions", Bulletin of the American Mathematical Society, 49: 270–277, doi:10.1090/S0002-9904-1943-07896-2, ISSN 0002-9904, MR 0007916
  • Parthasarathy, K. R. (1967). Probability measures on metric spaces. Probability and Mathematical Statistics. New York-London: Academic Press, Inc.

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.
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