Heine–Borel theorem

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In the topology of metric spaces the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states:

For a subset S of Euclidean space Rn, the following two statements are equivalent:

In the context of real analysis, the former property is sometimes used as the defining property of compactness. However, the two definitions cease to be equivalent when we consider subsets of more general metric spaces and in this generality only the latter property is used to define compactness. In fact, the Heine–Borel theorem for arbitrary metric spaces reads:

Contents

History and motivation

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. Central to the theory was the concept of uniform continuity and the theorem stating that every continuous function on a closed interval is uniformly continuous. 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. He used this proof in his 1862 lectures, which were published only in 1904. Later Eduard Heine, Karl Weierstrass and Salvatore Pincherle used similar techniques. Émile Borel in 1895 was the first to state and prove a form of what is now called the Heine–Borel theorem. His formulation was restricted to countable covers. Pierre Cousin (1895), Lebesgue (1898) and Schoenflies (1900) generalized it to arbitrary covers.[1]

Proof

If a set is compact, then it must be closed.

Let S be a subset of Rn. Observe first the following: if a is an limit point of S, then any finite collection C of open sets, such that each open set UC is disjoint from some neighborhood VU of a, fails to be a cover of S. Indeed, the intersection of the finite family of sets VU is a neighborhood W of a in Rn. Since a is a limit point of S, W must contain a point x in S. This xS is not covered by the family C, because every U in C is disjoint from VU and hence disjoint from W, which contains x.

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