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 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≤i≤n.
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.