Heine–Borel theorem

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. He used this proof in his 1852 lectures, which were published only in 1904.{{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 }} 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.{{cite arXiv |last=Sundström |first=Manya Raman | eprint=1006.4131v1 |title=A pedagogical history of compactness |class=math.HO |year=2010 }}

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

Let $S$ be a subset of $\backslash mathbb\{R\}^n$. 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 $U\backslash in\; C$ is disjoint from some neighborhood $V\_U$ 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 $\backslash mathbb\{R\}^n$. Since $a$ is a limit point of $S$, $W$ must contain a point $x$ in $S$. This $x\backslash in\; S$ 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 $a\backslash not\backslash in\; S$. Consider a collection $C\text{'}$ consisting of an open neighborhood $N(x)$ for each $x\backslash in\; S$, chosen small enough to not intersect some neighborhood $V\_x$ of $a$. Then $C\text{'}$ is an open cover of $S$, but any finite subcollection of $C\text{'}$ 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 $\backslash mathbb\{R\}^n$, and $U\_x$ a ball of radius 1 centered at $x\backslash in\backslash mathbb\{R\}^n$. Then the set of all such balls centered at $x\backslash in\; S$ is clearly an open cover of $S$, since $\backslash cup\_\{x\backslash in\; S\}\; U\_x$ 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 $p,q\backslash in\; S$, 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 $\backslash mathbb\{R\}^n$ and let $C\_K$ be an open cover of $K$. Then $U=\backslash mathbb\{R\}^n\backslash setminus\; K$ is an open set and

$$C\_T\; =\; C\_K\; \backslash cup\; \backslash \{U\backslash \}$$

is an open cover of $T$. Since $T$ is compact, then $C\_T$ has a finite subcover $C\_T\text{'}$, that also covers the smaller set $K$. Since $U$ does not contain any point of $K$, the set $K$ is already covered by $C\_K\text{'}\; =\; C\_T\text{'}\; \backslash setminus\; \backslash \{U\backslash \}$, 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 $\backslash mathbb\{R\}^n$ is bounded, then it can be enclosed within an $n$-box

$$T\_0\; =\; [-a,\; a]^n$$

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:

$$T\_0\; \backslash supset\; T\_1\; \backslash supset\; T\_2\; \backslash supset\; \backslash ldots\; \backslash supset\; T\_k\; \backslash supset\; \backslash ldots$$

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 $L\backslash in\; T\_k$ for each $k$.

Since $C$ covers $T\_0$, then it has some member $U\backslash in\; C$ such that $L\backslash in\; U$. Since $U$ is open, there is an $n$-ball $B(L)\backslash subseteq\; U$. For large enough $k$, one has $T\_k\backslash subseteq\; B(L)\backslash subseteq\; U$, 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).

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 precompactA set $S$ of a metric space $(X,\; d)$ is called precompact (or sometimes "totally bounded"), if for any $\backslash epsilon\; >\; 0$ there is a finite covering of $S$ by sets of diameter . and complete.A set $S$ of a metric space $(X,\; d)$ is called complete, if any Cauchy sequence in $S$ is convergent to a point in $S$.

The above follows directly from Jean Dieudonné, theorem 3.16.1,Diedonnné, Jean (1969): Foundations of Modern Analysis, Volume 1, enlarged and corrected printing. Academic Press, New York, London, p. 58 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;A point $x\backslash in\; X$ is said to be a cluster value of an infinite sequence $(x\_n)$ of elements of $x\_n\; \backslash in\; X$, if there exists a subsequence $(x\_\{n\_k\})$ such that $x\; =\; \backslash lim\_\{k\backslash to\backslash infty\}x\_\{n\_k\}$.
• (c) $X$ is precompact and complete.

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. These spaces are said to have the Heine–Borel property.

===In the theory of metric spaces===

A metric space $(X,d)$ is said to have the Heine–Borel property if each closed boundedA set $B$ in a metric space $(X,d)$ is said to be bounded if it is contained in a ball of a finite radius, i.e. there exists $a\backslash in\; X$ and $r>0$ such that $B\backslash subseteq\backslash \{x\backslash in\; X\backslash mid\; d(x,a)\backslash le\; r\backslash \}$. set in $X$ is compact.

Many metric spaces fail to have the Heine–Borel property, such as the metric space of rational numbers (or indeed any incomplete metric space). Complete metric spaces may also fail to have the property; for instance, no infinite-dimensional Banach spaces 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, $\backslash sigma$-compact, and locally compact.{{sfn|Williamson|Janos|1987}}

A topological vector space $X$ 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 boundedA set $B$ in a topological vector space $X$ is said to be bounded if for each neighborhood of zero $U$ in $X$ there exists a scalar $\backslash lambda$ such that $B\backslash subseteq\backslash lambda\backslash cdot\; U$. 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 $\{\backslash mathcal\; C\}^\backslash infty[0,1]$ of smooth functions on the interval $[0,1]$ with the metric $d(x,y)=\backslash sum\_\{k=0\}^\backslash infty\backslash frac\{1\}\{2^k\}\backslash cdot\backslash frac\{\backslash max\_\{t\backslash in[0,1]\}|x^\{(k)\}(t)-y^\{(k)\}(t)|\}\{1+\backslash max\_\{t\backslash in[0,1]\}|x^\{(k)\}(t)-y^\{(k)\}(t)|\}$ (here $x^\{(k)\}$ is the $k$-th derivative of the function $x\backslash in\; \{\backslash mathcal\; C\}^\backslash infty[0,1]$) has the Heine–Borel property as a topological vector space but not as a metric space. No infinite-dimensional Banach spaces have the Heine–Borel property (as topological vector spaces). But some infinite-dimensional Fréchet spaces do have, for instance, the space $C^\backslash infty(\backslash Omega)$ of smooth functions on an open set $\backslash Omega\backslash subset\backslash mathbb\{R\}^n${{sfn|Edwards|1965|loc=8.4.7}} and the space $H(\backslash Omega)$ of holomorphic functions on an open set $\backslash Omega\backslash subset\backslash mathbb\{C\}^n$.{{sfn|Edwards|1965|loc=8.4.7}} More generally, any quasi-complete nuclear space has the Heine–Borel property. All Montel spaces have the Heine–Borel property as well.

• Bolzano–Weierstrass theorem

{{reflist}}

• {{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 }}

• {{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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