In real analysis, a branch of mathematics, Cousin's theorem states that:

If for every point of a closed region (in modern terms, "closed and bounded") there is a circle of finite radius (in modern term, a "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.[1]

This result was originally proved by Pierre Cousin, a student of Henri Poincaré, in 1895, and it extends the original Heine–Borel theorem on compactness for arbitrary covers of compact subsets of . However, Pierre Cousin did not receive any credit. Cousin's theorem was generally attributed to Henri Lebesgue as the Borel–Lebesgue theorem. Lebesgue was aware of this result in 1898, and proved it in his 1903 dissertation.[1]

In modern terms, it is stated as:

Let 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 contains all subintervals of [a, b] which contains x and length smaller than δ. Then there exists a partition of non-overlapping intervals for [a, b], where and a=x0 < x1 < ⋯ < xn=b for all 1≤in.

Cousin's lemma is studied in reverse mathematics where it is one of the first third-order theorems that is hard to prove in terms of the comprehension axioms needed.

In Henstock–Kurzweil integration

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Cousin's theorem is instrumental in the study of Henstock–Kurzweil integration, and in this context, it is known as Cousin's lemma or the fineness theorem.

A gauge on   is a strictly positive real-valued function  , while a tagged partition of   is a finite sequence[2][3]

 

Given a gauge   and a tagged partition   of  , we say   is  -fine if for all  , we have  , where   denotes the open ball of radius   centred at  . Cousin's lemma is now stated as:

If  , then every gauge   has a  -fine partition.[4]

Proof of the theorem

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Cousin's theorem has an intuitionistic proof using the open induction principle, which reads as follows:

An open subset   of a closed real interval   is said to be inductive if it satisfies that   implies  . The open induction principle states that any inductive subset   of   must be the entire set.

Proof using open induction

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Let   be the set of points   such that there exists a  -fine tagged partition on   for some  . The set   is open, since it is downwards closed and any point in it is included in the open ray   for any associated partition.

Furthermore, it is inductive. For any  , suppose  . By that assumption (and using that either   or   to handle edge cases) we have a partition of length   with  . Then either   or  . In the first case  , so we can just replace   with   and get a partition of   that includes  .

If  , we may form a partition of length   that includes  . To show this, we split into the cases   or  . In the first case, we set  , in the second we set  . In both cases, we can set   and obtain a valid partition. So   in all cases, and   is inductive.

By open induction,  .

Notes

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  1. ^ a b Hildebrandt 1925, p. 29
  2. ^ Gordon, Russell (1994-08-01). The Integrals of Lebesgue, Denjoy, Perron, and Henstock. Graduate Studies in Mathematics. Vol. 4. Providence, Rhode Island: American Mathematical Society. doi:10.1090/gsm/004. ISBN 978-0-8218-3805-1.
  3. ^ Kurtz, Douglas S; Swartz, Charles W (October 2011). "Theories of Integration". Series in Real Analysis. 13. doi:10.1142/8291. ISBN 978-981-4368-99-5. ISSN 1793-1134.
  4. ^ Bartle 2001, p. 11

References

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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.
  • Raman, M. J. (1997). Understanding Compactness: A Historical Perspective, Master of Arts Thesis. University of California, Berkeley. arXiv:1006.4131.
  • Bartle, R. G. (2001). A Modern Theory of Integration, Graduate Studies in Mathematics 32, American Mathematical Society.