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| In [[real analysis]], a branch of mathematics, '''Cousin's theorem''' states that:
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| :If for every point of a closed region (in modern terms, "[[closed set|closed]] and [[bounded (set theory)|bounded]]") there is a circle of finite radius (in modern term, a "[[neighbourhood (mathematics)|neighborhood]]") , then the region can be divided into a finite number of subregions such that each subregion is interior to a circle of a given set having its center in the subregion.<ref name="h1">Hildebrandt 1925, p. 29</ref>
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| This result was proved and established by Pierre Cousin, a student of [[Henri Poincaré]], in 1895, and it is an extension of the original [[Heine–Borel theorem]] on [[compactness]] for arbitrary [[set cover|covers]] of any [[compact space|compact]] subsets of <math>\mathbb{R}^n</math>. However, Pierre Cousin did not receive any credit. '''Cousin's theorem''' was generally attributed to [[Henri Lebesgue]] and renamed as '''Borel–Lebesgue theorem''', who was aware of this result in 1898 and proved this in his dissertation in 1903.<ref name="h1"/>
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| Nowadays, it is stated as:
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| :Let <math>\mathcal{C}</math> be a full cover of [''a'', ''b''], that is, a collection of closed subintervals of [''a'', ''b''] with the property that for every ''x''∈[''a'', ''b''], there exists a ''δ''>0 so that <math>\mathcal{C}</math> contains all subintervals of [''a'', ''b''] which contains ''x'' and length smaller than ''δ''. Then there exists a partition {''I<sub>1</sub>'', ''I<sub>2</sub>'',...,''I<sub>n</sub>''} of non-overlapping intervals for [''a'', ''b''], where ''I<sub>i</sub>''=[''x<sub>i-1</sub>'', ''x<sub>i</sub>'']∈<math>\mathcal{C}</math> and ''a=x<sub>0</sub> < x<sub>1</sub> <...< x<sub>n</sub>=b'' for all ''1≤i≤n''. | |
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| Further, '''Cousin's theorem''' is mainly only used in [[Henstock–Kurzweil integral]] and is often called '''Fineness Theorem''' or [[Cousin's lemma]]. It can be stated as:
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| :If ''I'' := [''a'', ''b''] ⊆ '''R'''<sup>''n''</sup> is a [[degeneracy (mathematics)|nondegenerate]] [[compact space|compact]] interval and ''δ'' is any gauge defined on ''I'', then there always exists a tagged partition of ''I'' that is ''δ''-fine.<ref name="b1">Bartle 2001, p. 11</ref> | |
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| {{refbegin}}
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| *Hildebrandt, T. H. (1925). ''The Borel Theorem and its Generalizations'' In J. C. Abbott (Ed.), The Chauvenet Papers: A collection of Prize-Winning Expository Papers in Mathematics. Mathematical Association of America.
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| *Raman, M. J. (1997). ''Understanding Compactness: A Historical Perspective'', Master of Arts Thesis. University of California, Berkeley.
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| *Bartle, R. G. (2001). ''A Modern Theory of Integration'', Graduate Studies in Mathematics '''32''', American Mathematical Society.
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| {{refend}}
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| {{mathanalysis-stub}}
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| [[Category:Real analysis]]
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Hi also there. Let me begin with introducing the author, her name is Angele. Since I was 18 I've been working to be a financial expert. One of stuff she loves most is reading comics but she can't insure that it is her group. She's always loved living in florida and she loves a day living high. You can find my website here: http://www.medicalgroup-cerro.it/scarpe-hogan/
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