HOME

TheInfoList



OR:

In
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include co ...
, the Heine–Borel theorem, named after
Eduard Heine Heinrich Eduard Heine (16 March 1821 – 21 October 1881) was a German mathematician. Heine became known for results on special functions and in real analysis. In particular, he authored an important treatise on spherical harmonics and Leg ...
and
Émile Borel Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French people, French mathematician and politician. As a mathematician, he was known for his founding work in the areas of measure theory and probability. Biograp ...
, states: For a
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
S of
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
\mathbb^n, the following two statements are equivalent: *S is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
, that is, every open cover of S has a finite subcover *S is closed and bounded.


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 In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
and the theorem stating that every
continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
on a closed and bounded interval is uniformly continuous.
Peter Gustav Lejeune Dirichlet Johann Peter Gustav Lejeune Dirichlet (; ; 13 February 1805 – 5 May 1859) was a German mathematician. In number theory, he proved special cases of Fermat's last theorem and created analytic number theory. In analysis, he advanced the theory o ...
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 1852 lectures, which were published only in 1904. Later
Eduard Heine Heinrich Eduard Heine (16 March 1821 – 21 October 1881) was a German mathematician. Heine became known for results on special functions and in real analysis. In particular, he authored an important treatise on spherical harmonics and Leg ...
,
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (; ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the " father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a school t ...
and Salvatore Pincherle used similar techniques.
Émile Borel Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French people, French mathematician and politician. As a mathematician, he was known for his founding work in the areas of measure theory and probability. Biograp ...
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 In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
covers. Pierre Cousin (1895), Lebesgue (1898) and Schoenflies (1900) generalized it to arbitrary covers.


Proof

If a set is compact, then it must be closed. Let S be a subset of \mathbb^n. Observe first the following: if a is a
limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x contains a point of S other than x itself. A ...
of S, then any finite collection C of open sets, such that each open set U\in C is disjoint from some
neighborhood A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...
V_U of a, fails to be a cover of S. Indeed, the intersection of the finite family of sets V_U is a neighborhood W of a in \mathbb^n. Since a is a limit point of S, W must contain a point x in S. This x\in 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. 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\in S, chosen small enough to not intersect some neighborhood V_x 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. If a set is compact, then it is bounded. Let S be a compact set in \mathbb^n, and U_x a ball of radius 1 centered at x\in\mathbb^n. Then the set of all such balls centered at x\in S is clearly an open cover of S, since \cup_ U_x contains all of S. Since S 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 M be the maximum of the distances between them. Then if C_p and C_q are the centers (respectively) of unit balls containing arbitrary p,q\in S, the triangle inequality says: d(p, q)\le d(p, C_p) + d(C_p, C_q) + d(C_q, q)\le 1 + M + 1 = M + 2. So the diameter of S is bounded by M+2. Lemma: A closed subset of a compact set is compact. Let K be a closed subset of a compact set T in \mathbb^n and let C_K be an open cover of K. Then U=\mathbb^n\setminus K is an open set and C_T = C_K \cup \ is an open cover of T. Since T is compact, then C_T has a finite subcover C_T', that also covers the smaller set K. Since U does not contain any point of K, the set K is already covered by C_K' = C_T' \setminus \ , that is a finite subcollection of the original collection C_K. It is thus possible to extract from any open cover C_K of K a finite subcover. If a set is closed and bounded, then it is compact. If a set S in \mathbb^n is bounded, then it can be enclosed within an n-box T_0 = a, an where a>0. By the lemma above, it is enough to show that T_0 is compact. Assume, by way of contradiction, that T_0 is not compact. Then there exists an infinite open cover C of T_0 that does not admit any finite subcover. Through bisection of each of the sides of T_0, the box T_0 can be broken up into 2^n sub n-boxes, each of which has diameter equal to half the diameter of T_0. Then at least one of the 2^n sections of T_0 must require an infinite subcover of C, otherwise C itself would have a finite subcover, by uniting together the finite covers of the sections. Call this section T_1. Likewise, the sides of T_1 can be bisected, yielding 2^n sections of T_1, at least one of which must require an infinite subcover of C. Continuing in like manner yields a decreasing sequence of nested n-boxes: T_0 \supset T_1 \supset T_2 \supset \ldots \supset T_k \supset \ldots where the side length of T_k is (2a)/2^k, which tends to 0 as k tends to infinity. Let us define a sequence (x_k) such that each x_k is in T_k. This sequence is Cauchy, so it must converge to some limit L. Since each T_k is closed, and for each k the sequence (x_k) is eventually always inside T_k, we see that L\in T_k for each k. Since C covers T_0, then it has some member U\in C such that L\in U. Since U is open, there is an n-ball B(L)\subseteq U. For large enough k, one has T_k\subseteq B(L)\subseteq U, but then the infinite number of members of C needed to cover T_k can be replaced by just one: U, a contradiction. Thus, T_0 is compact. Since S is closed and a subset of the compact set T_0, then S is also compact (see the lemma above).


Generalization of the Heine-Borel theorem

In general metric spaces, we have the following theorem: For a subset S of a metric space (X, d), the following two statements are equivalent: * S is compact, * S is precompact and complete. The above follows directly from Jean Dieudonné, theorem 3.16.1, which states: For a metric space (X, d), the following three conditions are equivalent: * (a) X is compact; * (b) any infinite sequence in X has at least a cluster value; * (c) X is precompact and complete.


Heine–Borel property

The Heine–Borel theorem does not hold as stated for general
metric Metric or metrical may refer to: Measuring * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics ...
and
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
s, and this gives rise to the necessity to consider special classes of spaces where this proposition is true. These spaces are said to have the Heine–Borel property.


In the theory of metric spaces

A
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
(X,d) is said to have the Heine–Borel property if each closed bounded set in X is compact. Many metric spaces fail to have the Heine–Borel property, such as the metric space of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
s (or indeed any incomplete metric space). Complete metric spaces may also fail to have the property; for instance, no infinite-dimensional
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
s have the Heine–Borel property (as metric spaces). Even more trivially, if the real line is not endowed with the usual metric, it may fail to have the Heine–Borel property. A metric space (X,d) has a Heine–Borel metric which is Cauchy locally identical to d if and only if it is complete, \sigma-compact, and
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...
.


In the theory of topological vector spaces

A
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
X is said to have the Heine–Borel property (R.E. Edwards uses the term ''boundedly compact space'') if each closed bounded set in X is compact.In the case when the topology of a topological vector space X is generated by some metric d this definition is not equivalent to the definition of the Heine–Borel property of X as a metric space, since the notion of bounded set in X as a metric space is different from the notion of bounded set in X as a topological vector space. For instance, the space ^\infty ,1/math> of smooth functions on the interval ,1/math> with the metric d(x,y)=\sum_^\infty\frac\cdot\frac (here x^ is the k-th derivative of the function x\in ^\infty ,1/math>) has the Heine–Borel property as a topological vector space but not as a metric space. No infinite-dimensional
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
s have the Heine–Borel property (as topological vector spaces). But some infinite-dimensional
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to ...
s do have, for instance, the space C^\infty(\Omega) of smooth functions on an open set \Omega\subset\mathbb^n and the space H(\Omega) of holomorphic functions on an open set \Omega\subset\mathbb^n. More generally, any quasi-complete nuclear space has the Heine–Borel property. All
Montel space In functional analysis and related areas of mathematics, a Montel space, named after Paul Montel, is any topological vector space (TVS) in which an analog of Montel's theorem holds. Specifically, a Montel space is a barrelled topological vector ...
s have the Heine–Borel property as well.


See also

*
Bolzano–Weierstrass theorem In mathematics, specifically in real analysis, the Bolzano–Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space \R^n. The theorem states that ea ...


Notes


References

* * BookOfProofs
Heine-Borel Property
* * * *


External links

* *
"An Analysis of the First Proofs of the Heine-Borel Theorem - Lebesgue's Proof"
{{DEFAULTSORT:Heine-Borel Theorem Theorems in real analysis General topology Properties of topological spaces Compactness theorems Articles containing proofs