Editing Heine–Borel theorem

{{short description|Subset of Euclidean space is compact if and only if it is closed and bounded}}
In [[real analysis]], the '''Heine–Borel theorem''', named after [[Eduard Heine]] and [[Émile Borel]], states:

For a [[subset]] <math>S</math> of [[Euclidean space]] <math>\mathbb{R}^n</math>, the following two statements are equivalent:
*<math>S</math> is [[compact space#Open cover definition|compact]], that is, every open [[cover (topology)|cover]] of <math>S</math> has a finite subcover
*<math>S</math> is [[closed set|closed]] and [[bounded set|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]] and the theorem stating that every [[continuous function]] on a closed and bounded interval is uniformly continuous. [[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.<ref name="Sundström"/> He used this proof in his 1852 lectures, which were published only in 1904.<ref name="Sundström">{{cite journal | journal = [[American Mathematical Monthly]] | title = A Pedagogical History of Compactness | last1 = Raman-Sundström | first1 = Manya | date = August–September 2015 | volume = 122 | issue = 7 | pages = 619–635 | jstor = 10.4169/amer.math.monthly.122.7.619| doi = 10.4169/amer.math.monthly.122.7.619 | arxiv = 1006.4131 | s2cid = 119936587 }}</ref> 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 set|countable]] covers. Pierre Cousin (1895), [[Lebesgue]] (1898) and [[Arthur Schoenflies|Schoenflies]] (1900) generalized it to arbitrary covers.<ref name="sundstrom_2010">{{cite arXiv |last=Sundström |first=Manya Raman | eprint=1006.4131v1 |title=A pedagogical history of compactness |class=math.HO |year=2010 }}</ref>

== Proof ==

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

Let <math>S</math> be a subset of <math>\mathbb{R}^n</math>.  Observe first the following: if <math>a</math> is a [[limit point]] of <math>S</math>, then any finite collection <math>C</math> of open sets, such that each open set <math>U\in C</math> is disjoint from some [[Neighbourhood (mathematics)#Definitions|neighborhood]] <math>V_U</math> of <math>a</math>, fails to be a cover of <math>S</math>. Indeed, the intersection of the finite family of sets <math>V_U</math> is a neighborhood <math>W</math> of <math>a</math> in <math>\mathbb{R}^n</math>. Since <math>a</math> is a limit point of <math>S</math>, <math>W</math> must contain a point <math>x</math> in <math>S</math>. This <math>x\in S</math> is not covered by the family <math>C</math>, because every <math>U</math> in <math>C</math> is disjoint from <math>V_U</math> and hence disjoint from <math>W</math>, which contains <math>x</math>.

If <math>S</math> is compact but not closed, then it has a limit point <math>a\not\in S</math>. Consider a collection <math>C'</math> consisting of an open neighborhood <math>N(x)</math> for each <math>x\in S</math>, chosen small enough to not intersect some neighborhood <math>V_x</math> of <math>a</math>.  Then <math>C'</math> is an open cover of <math>S</math>, but any finite subcollection of <math>C'</math> has the form of <math>C</math> discussed previously, and thus cannot be an open subcover of <math>S</math>. This contradicts the compactness of <math>S</math>.  Hence, every limit point of <math>S</math> is in <math>S</math>, so <math>S</math> is closed.

The proof above applies with almost no change to showing that any compact subset <math>S</math> of a [[Hausdorff space|Hausdorff]] topological space <math>X</math> is closed in <math>X</math>.

'''If a set is compact, then it is bounded.'''

Let <math>S</math> be a compact set in <math>\mathbb{R}^n</math>, and <math>U_x</math> a ball of radius 1 centered at <math>x\in\mathbb{R}^n</math>. Then the set of all such balls centered at <math>x\in S</math> is clearly an open cover of <math>S</math>, since <math>\cup_{x\in S} U_x</math> contains all of <math>S</math>. Since <math>S</math> 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 <math>M</math> be the maximum of the distances between them. Then if <math>C_p</math> and <math>C_q</math> are the centers (respectively) of unit balls containing arbitrary <math>p,q\in S</math>, the triangle inequality says:

<math display="block">
d(p, q)\le d(p, C_p) + d(C_p, C_q) + d(C_q, q)\le 1 + M + 1 = M + 2.
</math>

So the diameter of <math>S</math> is bounded by <math>M+2</math>.

'''Lemma: A closed subset of a compact set is compact.'''

Let <math>K</math> be a closed subset of a compact set <math>T</math> in <math>\mathbb{R}^n</math> and let <math>C_K</math> be an open cover of <math>K</math>. Then <math>U=\mathbb{R}^n\setminus K</math> is an open set and
 
<math display="block"> C_T = C_K \cup \{U\} </math>

is an open cover of <math>T</math>. Since <math>T</math> is compact, then <math>C_T</math> has a finite subcover <math> C_T'</math>, that also covers the smaller set <math>K</math>.  Since <math>U</math> does not contain any point of <math>K</math>, the set <math>K</math> is already covered by <math> C_K' = C_T' \setminus \{U\} </math>, that is a finite subcollection of the original collection <math>C_K</math>.  It is thus possible to extract from any open cover <math>C_K</math> of <math>K</math> a finite subcover.

'''If a set is closed and bounded, then it is compact.'''

If a set <math>S</math> in <math>\mathbb{R}^n</math> is bounded, then it can be enclosed within an <math>n</math>-box

<math display="block"> T_0 = [-a, a]^n</math>

where <math>a>0</math>. By the lemma above, it is enough to show that <math>T_0</math> is compact.

Assume, by way of contradiction, that <math>T_0</math> is not compact. Then there exists an infinite open cover <math>C</math> of <math>T_0</math> that does not admit any finite subcover. Through bisection of each of the sides of <math>T_0</math>, the box <math>T_0</math> can be broken up into <math>2^n</math> sub <math>n</math>-boxes, each of which has diameter equal to half the diameter of <math>T_0</math>. Then at least one of the <math>2^n</math> sections of <math>T_0</math> must require an infinite subcover of <math>C</math>, otherwise <math>C</math> itself would have a finite subcover, by uniting together the finite covers of the sections. Call this section <math>T_1</math>.

Likewise, the sides of <math>T_1</math> can be bisected, yielding <math>2^n</math> sections of <math>T_1</math>, at least one of which must require an infinite subcover of <math>C</math>. Continuing in like manner yields a decreasing sequence of nested <math>n</math>-boxes:

<math display="block"> T_0 \supset T_1 \supset T_2 \supset \ldots \supset T_k \supset \ldots </math>

where the side length of <math>T_k</math> is <math>(2a)/2^k</math>, which tends to 0 as <math>k</math> tends to infinity. Let us define a sequence <math>(x_k)</math> such that each <math>x_k</math> is in <math>T_k</math>. This sequence is [[Cauchy_sequence | Cauchy]], so it must converge to some limit <math>L</math>. Since each <math>T_k</math> is closed, and for each <math>k</math> the sequence <math>(x_k)</math> is eventually always inside <math>T_k</math>, we see that <math>L\in T_k</math> for each <math>k</math>.

Since <math>C</math> covers <math>T_0</math>, then it has some member <math>U\in C</math> such that <math>L\in U</math>. Since <math>U</math> is open, there is an <math>n</math>-ball <math>B(L)\subseteq U</math>. For large enough <math>k</math>, one has <math>T_k\subseteq B(L)\subseteq U</math>, but then the infinite number of members of <math>C</math> needed to cover <math>T_k</math> can be replaced by just one: <math>U</math>, a contradiction.

Thus, <math>T_0</math> is compact. Since <math>S</math> is closed and a subset of the compact set <math>T_0</math>, then <math>S</math> 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 <math>S</math> of a metric space <math>(X, d)</math>, the following two statements are equivalent:
* <math>S</math> is compact,
* <math>S</math> is precompact<ref>A set <math>S</math> of a metric space <math>(X, d)</math> is called precompact (or sometimes "totally bounded"), if for any <math>\epsilon > 0</math> there is a finite covering of <math>S</math> by sets of diameter <math>< \epsilon</math>.</ref> and complete.<ref>A set <math>S</math> of a metric space <math>(X, d)</math> is called complete, if any [[Cauchy sequence#In a metric space|Cauchy sequence]] in <math>S</math> is convergent to a point in <math>S</math>.</ref>

The above follows directly from [[Jean Dieudonné]], theorem 3.16.1,<ref>Diedonnné, Jean (1969): Foundations of Modern Analysis, Volume 1, enlarged and corrected printing. Academic Press, New York, London, p.&nbsp;58</ref> which states:

For a metric space <math>(X, d)</math>, the following three conditions are equivalent:
* (a) <math>X</math> is compact;
* (b) any infinite sequence in <math>X</math> has at least a cluster value;<ref>A point <math>x\in X</math> is said to be a cluster value of an infinite sequence <math>(x_n)</math> of elements of <math>x_n \in X</math>, if there exists a subsequence <math>(x_{n_k})</math> such that <math>x = \lim_{k\to\infty}x_{n_k}</math>.</ref>
* (c) <math>X</math> is precompact and complete.

== Heine–Borel property ==
The Heine–Borel theorem does not hold as stated for general [[Metric space|metric]] and [[topological vector space]]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]] <math>(X,d)</math> is said to have the '''Heine–Borel property''' if each closed bounded<ref>A set <math>B</math> in a metric space <math>(X,d)</math> is said to be ''bounded'' if it is contained in a ball of a finite radius, i.e. there exists <math>a\in X</math> and <math>r>0</math> such that <math>B\subseteq\{x\in X\mid d(x,a)\le r\}</math>.</ref> set in <math>X</math> is compact.

Many metric spaces fail to have the Heine–Borel property, such as the metric space of [[rational number]]s (or indeed any incomplete metric space). Complete metric spaces may also fail to have the property; for instance, no infinite-dimensional [[Banach space]]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 <math>(X,d)</math> has a Heine–Borel metric which is Cauchy locally identical to <math>d</math> if and only if it is [[complete space|complete]], [[sigma-compact|<math>\sigma</math>-compact]], and [[locally compact space|locally compact]].{{sfn|Williamson|Janos|1987}}

===In the theory of topological vector spaces===
A [[topological vector space]] <math>X</math> is said to have the '''Heine–Borel property'''{{sfn|Kirillov|Gvishiani|1982|loc=Theorem 28}} (R.E. Edwards uses the term ''boundedly compact space''{{sfn|Edwards|1965|loc=8.4.7}}) if each closed bounded<ref>A set <math>B</math> in a topological vector space <math>X</math> is said to be ''bounded'' if for each neighborhood of zero <math>U</math> in <math>X</math> there exists a scalar <math>\lambda</math> such that <math>B\subseteq\lambda\cdot U</math>.</ref> set in <math>X</math> is compact.<ref>In the case when the topology of a topological vector space <math>X</math> is generated by some metric <math>d</math> this definition is not equivalent to the definition of the Heine–Borel property of <math>X</math> as a metric space, since the notion of bounded set in <math>X</math> as a metric space is different from the notion of bounded set in <math>X</math> as a topological vector space. For instance, the space <math>{\mathcal C}^\infty[0,1]</math> of smooth functions on the interval <math>[0,1]</math> with the metric <math>d(x,y)=\sum_{k=0}^\infty\frac{1}{2^k}\cdot\frac{\max_{t\in[0,1]}|x^{(k)}(t)-y^{(k)}(t)|}{1+\max_{t\in[0,1]}|x^{(k)}(t)-y^{(k)}(t)|}</math> (here <math>x^{(k)}</math> is the <math>k</math>-th derivative of the function <math>x\in {\mathcal C}^\infty[0,1]</math>) has the Heine–Borel property as a topological vector space but not as a metric space.</ref> No infinite-dimensional [[Banach space]]s have the Heine–Borel property (as topological vector spaces). But some infinite-dimensional [[Fréchet space]]s do have, for instance, the space <math>C^\infty(\Omega)</math> of smooth functions on an open set <math>\Omega\subset\mathbb{R}^n</math>{{sfn|Edwards|1965|loc=8.4.7}} and the space <math>H(\Omega)</math> of holomorphic functions on an open set <math>\Omega\subset\mathbb{C}^n</math>.{{sfn|Edwards|1965|loc=8.4.7}}  More generally, any quasi-complete [[nuclear space]] has the Heine–Borel property. All [[Montel space]]s have the Heine–Borel property as well.

== See also ==
* [[Bolzano–Weierstrass theorem]]

==Notes==
{{reflist}}

== References ==
* {{cite journal | author=P. Dugac | title=Sur la correspondance de Borel et le théorème de Dirichlet–Heine–Weierstrass–Borel–Schoenflies–Lebesgue | journal= Arch. Int. Hist. Sci. | year=1989 | volume=39 | pages=69–110}}
* BookOfProofs: [http://www.bookofproofs.org/branches/heine-borel-property-defines-compact-subsets/ Heine-Borel Property]
*{{cite book | last1=Jeffreys| first1=H. |last2=Jeffreys| first2=B.S. | title=Methods of Mathematical Physics | url=https://archive.org/details/methodsofmathema0000jeff| url-access=registration| publisher=Cambridge University Press | year=1988 | isbn=978-0521097239 }}
* {{cite journal | last1=Williamson | first1=R.| last2=Janos| first2=L. | title=Constructing metrics with the Heine-Borel property | journal= Proc. AMS | year=1987 | volume=100 | issue=3| pages=567–573| doi=10.1090/S0002-9939-1987-0891165-X| doi-access=free}}
*{{cite book | last1=Kirillov| first1=A.A. | last2=Gvishiani |first2=A.D. | title=Theorems and Problems in Functional Analysis | publisher=Springer-Verlag New York | year=1982 | isbn=978-1-4613-8155-6}}
*{{cite book | last=Edwards| first=R.E. | title=Functional analysis | publisher=Holt, Rinehart and Winston | year=1965 | isbn=0030505356 }}

== External links ==
* {{cite video|people=Ivan Kenig, Dr. Prof. Hans-Christian Graf v. Botthmer, Dmitrij Tiessen, Andreas Timm, Viktor Wittman |title=The Heine–Borel Theorem |format=avi • mp4 • mov • swf • streamed video |date=2004 |publisher=Leibniz Universität |location=Hannover |url=http://www.math.uni-sb.de/ag/schreyer/oliver/calendar.algebraicsurface.net/calendar.php?mode=youTube&day=07 |url-status=dead |archive-url=https://web.archive.org/web/20110719111010/http://www.math.uni-sb.de/ag/schreyer/oliver/calendar.algebraicsurface.net/calendar.php?mode=youTube&day=07 |archive-date=2011-07-19 }}
* {{springer|title=Borel-Lebesgue covering theorem|id=p/b017100}}
* [http://mathworld.wolfram.com/Heine-BorelTheorem.html Mathworld "Heine-Borel Theorem"]
*[https://old.maa.org/press/periodicals/convergence/an-analysis-of-the-first-proofs-of-the-heine-borel-theorem-lebesgues-proof "An Analysis of the First Proofs of the Heine-Borel Theorem - Lebesgue's Proof"]

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