Federer–Morse theorem: Difference between revisions
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{{distinguish|Morse–Sard–Federer theorem}} |
{{distinguish|Morse–Sard–Federer theorem}} |
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In mathematics, the '''Federer–Morse theorem''', introduced by {{harvs|txt|author2-link=Anthony Morse|last2=Morse|author1-link=Herbert Federer|last1=Federer|year=1943}}, states that if ''f'' is a [[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 '' |
In mathematics, the '''Federer–Morse theorem''', introduced by {{harvs|txt|author2-link=Anthony Morse|last2=Morse|author1-link=Herbert Federer|last1=Federer|year=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''. |
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Moreover, the inverse of that restriction is a Borel [[Section_(category_theory)|section]] of ''f''. |
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==See also== |
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* [[Uniformization_(set_theory)|Uniformization]] |
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==References== |
==References== |
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Revision as of 09:31, 26 September 2014
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. Moreover, the inverse of that restriction is a Borel section of f.
See also
References
- 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