![]() ![]() If S is compact but not closed, then it has a limit point a not in S. This x ∈ S is not covered by the family C, because every U in C is disjoint from V U and hence disjoint from W, which contains x. Since a is a limit point of S, W must contain a point x in S. Indeed, the intersection of the finite family of sets V U is a neighborhood W of a in R n. Observe first the following: if a is a limit point of S, then any finite collection C of open sets, such that each open set U ∈ C is disjoint from some neighborhood V U of a, fails to be a cover of S. ![]() If a set is compact, then it must be closed. Pierre Cousin (1895), Lebesgue (1898) and Schoenflies (1900) generalized it to arbitrary covers. His formulation was restricted to countable covers. Émile Borel in 1895 was the first to state and prove a form of what is now called the Heine–Borel theorem. Later Eduard Heine, Karl Weierstrass and Salvatore Pincherle used similar techniques. He used this proof in his 1852 lectures, which were published only in 1904. Peter Gustav Lejeune Dirichlet was the first to prove this and implicitly he used the existence of a finite subcover of a given open cover of a closed interval in his proof. Central to the theory was the concept of uniform continuity and the theorem stating that every continuous function on a closed and bounded interval is uniformly continuous. The history of what today is called the Heine–Borel theorem starts in the 19th century, with the search for solid foundations of real analysis. S is compact, that is, every open cover of S has a finite subcover.In real analysis the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states:įor a subset S of Euclidean space R n, the following two statements are equivalent: ![]() Subset of Euclidean space is compact if and only if it is closed and bounded ![]()
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