Cousin's theorem

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 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 covers of any compact subsets of $\mathbb{R}^n$. 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.^[1]

Nowadays, it is stated as:

Let $\mathcal{C}$ 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 $\mathcal{C}$ contains all subintervals of [a, b] which contains x and length smaller than δ. Then there exists a partition {I₁, I₂,...,I_n} of non-overlapping intervals for [a, b], where I_i=[x_i-1, x_i]∈$\mathcal{C}$ and a=x₀ < x₁ <...< x_n=b for all 1≤i≤n.

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:

If I := [a, b] ⊆ Rⁿ is a nondegenerate compact interval and δ is any gauge defined on I, then there always exists a tagged partition of I that is δ-fine.^[2]

Notes

1. Hildebrandt 1925, p. 29
2. Bartle 2001, p. 11

References

• 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.
• Bartle, R. G. (2001). A Modern Theory of Integration, Graduate Studies in Mathematics 32, American Mathematical Society.