In real analysis 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: 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.
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.
[1] 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.
Pierre Cousin (1895), Lebesgue (1898) and Schoenflies (1900) generalized it to arbitrary covers.
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 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.
If S is compact but not closed, then it has a limit point a not in S. Consider a collection C′ consisting of an open neighborhood N(x) for each x ∈ S, chosen small enough to not intersect some neighborhood Vx of a.
Then C′ is an open cover of S, but any finite subcollection of C′ has the form of C discussed previously, and thus cannot be an open subcover of S. This contradicts the compactness of S. Hence, every limit point of S is in S, so S is closed.
The proof above applies with almost no change to showing that any compact subset S of a Hausdorff topological space X is closed in X.
is compact, take a finite subcover of this cover.
This subcover is the finite union of balls of radius 1.
Consider all pairs of centers of these (finitely many) balls (of radius 1) and let
are the centers (respectively) of unit balls containing arbitrary
Let K be a closed subset of a compact set T in Rn and let CK be an open cover of K. Then U = Rn \ K is an open set and
is an open cover of T. Since T is compact, then CT has a finite subcover
that is a finite subcollection of the original collection CK.
It is thus possible to extract from any open cover CK of K a finite subcover.
If a set is closed and bounded, then it is compact.
If a set S in Rn is bounded, then it can be enclosed within an n-box
Then there exists an infinite open cover C of T0 that does not admit any finite subcover.
Then at least one of the 2n sections of T0 must require an infinite subcover of C, otherwise C itself would have a finite subcover, by uniting together the finite covers of the sections.
Likewise, the sides of T1 can be bisected, yielding 2n sections of T1, at least one of which must require an infinite subcover of C. Continuing in like manner yields a decreasing sequence of nested n-boxes:
where the side length of Tk is (2 a) / 2k, which tends to 0 as k tends to infinity.
For large enough k, one has Tk ⊆ B(L) ⊆ U, but then the infinite number of members of C needed to cover Tk can be replaced by just one: U, a contradiction.
In general metric spaces, we have the following theorem: For a subset
, the following two statements are equivalent: The above follows directly from Jean Dieudonné, theorem 3.16.1,[5] which states: For a metric space
, the following three conditions are equivalent: The Heine–Borel theorem does not hold as stated for general metric and topological vector spaces, and this gives rise to the necessity to consider special classes of spaces where this proposition is true.
Even more trivially, if the real line is not endowed with the usual metric, it may fail to have the Heine–Borel property.
of smooth functions on an open set
[10] More generally, any quasi-complete nuclear space has the Heine–Borel property.