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 ...
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 ...
, the following two statements are equivalent:
*
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
has a finite subcover
*
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
be a subset of
. Observe first the following: if
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
, then any finite collection
of open sets, such that each open set
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 ...
of
, fails to be a cover of
. Indeed, the intersection of the finite family of sets
is a neighborhood
of
in
. Since
is a limit point of
,
must contain a point
in
. This
is not covered by the family
, because every
in
is disjoint from
and hence disjoint from
, which contains
.
If
is compact but not closed, then it has a limit point
. Consider a collection
consisting of an open neighborhood
for each
, chosen small enough to not intersect some neighborhood
of
. Then
is an open cover of
, but any finite subcollection of
has the form of
discussed previously, and thus cannot be an open subcover of
. This contradicts the compactness of
. Hence, every limit point of
is in
, so
is closed.
The proof above applies with almost no change to showing that any compact subset
of a
Hausdorff topological space
is closed in
.
If a set is compact, then it is bounded.
Let
be a compact set in
, and
a ball of radius 1 centered at
. Then the set of all such balls centered at
is clearly an open cover of
, since
contains all of
. Since
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
be the maximum of the distances between them. Then if
and
are the centers (respectively) of unit balls containing arbitrary
, the triangle inequality says:
So the diameter of
is bounded by
.
Lemma: A closed subset of a compact set is compact.
Let
be a closed subset of a compact set
in
and let
be an open cover of
. Then
is an open set and
is an open cover of
. Since
is compact, then
has a finite subcover
, that also covers the smaller set
. Since
does not contain any point of
, the set
is already covered by
, that is a finite subcollection of the original collection
. It is thus possible to extract from any open cover
of
a finite subcover.
If a set is closed and bounded, then it is compact.
If a set
in
is bounded, then it can be enclosed within an
-box
where
. By the lemma above, it is enough to show that
is compact.
Assume, by way of contradiction, that
is not compact. Then there exists an infinite open cover
of
that does not admit any finite subcover. Through bisection of each of the sides of
, the box
can be broken up into
sub
-boxes, each of which has diameter equal to half the diameter of
. Then at least one of the
sections of
must require an infinite subcover of
, otherwise
itself would have a finite subcover, by uniting together the finite covers of the sections. Call this section
.
Likewise, the sides of
can be bisected, yielding
sections of
, at least one of which must require an infinite subcover of
. Continuing in like manner yields a decreasing sequence of nested
-boxes:
where the side length of
is
, which tends to 0 as
tends to infinity. Let us define a sequence
such that each
is in
. This sequence is
Cauchy, so it must converge to some limit
. Since each
is closed, and for each
the sequence
is eventually always inside
, we see that
for each
.
Since
covers
, then it has some member
such that
. Since
is open, there is an
-ball
. For large enough
, one has
, but then the infinite number of members of
needed to cover
can be replaced by just one:
, a contradiction.
Thus,
is compact. Since
is closed and a subset of the compact set
, then
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
of a metric space
, the following two statements are equivalent:
*
is compact,
*
is precompact and complete.
The above follows directly from
Jean Dieudonné, theorem 3.16.1, which states:
For a metric space
, the following three conditions are equivalent:
* (a)
is compact;
* (b) any infinite sequence in
has at least a cluster value;
* (c)
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 ...
is said to have the Heine–Borel property if each closed bounded set in
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
has a Heine–Borel metric which is Cauchy locally identical to
if and only if it is
complete,
-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 ...
is said to have the Heine–Borel property (R.E. Edwards uses the term ''boundedly compact space'') if each closed bounded set in
is compact.
[In the case when the topology of a topological vector space is generated by some metric this definition is not equivalent to the definition of the Heine–Borel property of as a metric space, since the notion of bounded set in as a metric space is different from the notion of bounded set in as a topological vector space. For instance, the 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