Misplaced Pages

#670329

33-498: For a subset S of Euclidean space R , 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. 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

66-440: 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 sub n -boxes, each of which has diameter equal to half the diameter of T 0 . Then at least one of

99-416: A set A is a subset of a set B if all 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 are unequal, then A is a proper subset of B . The relationship of one set being a subset of another is called inclusion (or sometimes containment ). A is a subset of B may also be expressed as B includes (or contains) A or A

132-405: 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 R . Since a is a limit point of S , W must contain a point x in S . This x ∈ 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

165-526: A Heine–Borel metric which is Cauchy locally identical to d {\displaystyle d} if and only if it is complete , σ {\displaystyle \sigma } -compact , and locally compact . A topological vector space X {\displaystyle 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 {\displaystyle X}

198-646: A ball of radius 1 centered at x ∈ R n {\displaystyle x\in \mathbf {R} ^{n}} . Then the set of all such balls centered at x ∈ S {\displaystyle x\in S} is clearly an open cover of S {\displaystyle S} , since ∪ x ∈ S U x {\displaystyle \cup _{x\in S}U_{x}} contains all of S {\displaystyle S} . Since S {\displaystyle S}

231-434: 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 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

264-911: A proof technique known as the element argument : Let sets A and B be given. To prove that A ⊆ B , {\displaystyle A\subseteq B,} The validity of this technique can be seen as a consequence of universal generalization : the technique shows ( c ∈ A ) ⇒ ( c ∈ B ) {\displaystyle (c\in A)\Rightarrow (c\in B)} for an arbitrarily chosen element c . Universal generalisation then implies ∀ x ( x ∈ A ⇒ x ∈ B ) , {\displaystyle \forall x\left(x\in A\Rightarrow x\in B\right),} which

297-514: Is less than y (an irreflexive relation ). Similarly, using the convention that ⊂ {\displaystyle \subset } is proper subset, if A ⊆ B , {\displaystyle A\subseteq B,} then A may or may not equal B , but if A ⊂ B , {\displaystyle A\subset B,} then A definitely does not equal B . Another example in an Euler diagram : The set of all subsets of S {\displaystyle S}

330-486: Is vacuously a subset of any set X . Some authors use the symbols ⊂ {\displaystyle \subset } and ⊃ {\displaystyle \supset } to indicate subset and superset respectively; that is, with the same meaning as and instead of the symbols ⊆ {\displaystyle \subseteq } and ⊇ . {\displaystyle \supseteq .} For example, for these authors, it

363-425: 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 R is bounded, then it can be enclosed within an n -box T 0 = [ − a , a ] n {\displaystyle T_{0}=[-a,a]^{n}} where

SECTION 10

#1732869961671

396-457: Is an open cover of T . Since T is compact, then C T has a finite subcover C T ′ , {\displaystyle 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 ′ ∖ { U } , {\displaystyle C_{K}'=C_{T}'\setminus \{U\},} that

429-803: Is called its power set , and is denoted by P ( S ) {\displaystyle {\mathcal {P}}(S)} . The inclusion relation ⊆ {\displaystyle \subseteq } is a partial order on the set P ( S ) {\displaystyle {\mathcal {P}}(S)} defined by A ≤ B ⟺ A ⊆ B {\displaystyle A\leq B\iff A\subseteq B} . We may also partially order P ( S ) {\displaystyle {\mathcal {P}}(S)} by reverse set inclusion by defining A ≤ B  if and only if  B ⊆ A . {\displaystyle A\leq B{\text{ if and only if }}B\subseteq A.} For

462-422: 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 {\displaystyle M} be the maximum of the distances between them. Then if C p {\displaystyle C_{p}} and C q {\displaystyle C_{q}} are

495-460: Is compact. 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 ∞ ( Ω ) {\displaystyle C^{\infty }(\Omega )} of smooth functions on an open set Ω ⊂ R n {\displaystyle \Omega \subset \mathbb {R} ^{n}} and

528-533: Is equivalent to A ⊆ B , {\displaystyle A\subseteq B,} as stated above. If A and B are sets and every element of A is also an element of B , then: If A is a subset of B , but A is not equal to B (i.e. there exists at least one element of B which is not an element of A ), then: The empty set , written { } {\displaystyle \{\}} or ∅ , {\displaystyle \varnothing ,} has no elements, and therefore

561-427: 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 {\displaystyle S} be a compact set in R n {\displaystyle \mathbf {R} ^{n}} , and U x {\displaystyle U_{x}}

594-498: Is included (or contained) in B . A k -subset is a subset with k elements. When quantified, A ⊆ B {\displaystyle A\subseteq B} is represented as ∀ x ( x ∈ A ⇒ x ∈ B ) . {\displaystyle \forall x\left(x\in A\Rightarrow x\in B\right).} One can prove the statement A ⊆ B {\displaystyle A\subseteq B} by applying

627-408: Is open, there is an n -ball B ( L ) ⊆ U . For large enough k , one has T k ⊆ B ( L ) ⊆ 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

660-405: Is true of every set A that A ⊂ A . {\displaystyle A\subset A.} (a reflexive relation ). Other authors prefer to use the symbols ⊂ {\displaystyle \subset } and ⊃ {\displaystyle \supset } to indicate proper (also called strict) subset and proper superset respectively; that is, with

693-500: The k -tuple from { 0 , 1 } k , {\displaystyle \{0,1\}^{k},} of which the i th coordinate is 1 if and only if s i {\displaystyle s_{i}} is a member of T . The set of all k {\displaystyle k} -subsets of A {\displaystyle A} is denoted by ( A k ) {\displaystyle {\tbinom {A}{k}}} , in analogue with

SECTION 20

#1732869961671

726-715: The 2 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 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 ⊃ T 1 ⊃ T 2 ⊃ … ⊃ T k ⊃ … {\displaystyle T_{0}\supset T_{1}\supset T_{2}\supset \ldots \supset T_{k}\supset \ldots } where

759-536: The centers (respectively) of unit balls containing arbitrary p , q ∈ S {\displaystyle p,q\in S} , the triangle inequality says: d ( p , q ) ≤ d ( p , C p ) + d ( C p , C q ) + d ( C q , q ) ≤ 1 + M + 1 = M + 2. {\displaystyle d(p,q)\leq d(p,C_{p})+d(C_{p},C_{q})+d(C_{q},q)\leq 1+M+1=M+2.} So

792-458: The diameter of S {\displaystyle S} is bounded by M + 2 {\displaystyle M+2} . Lemma: A closed subset of a compact set is compact. Let K be a closed subset of a compact set T in R and let C K be an open cover of K . Then U = R \ K is an open set and C T = C K ∪ { U } {\displaystyle C_{T}=C_{K}\cup \{U\}}

825-565: The following theorem: For a subset S {\displaystyle S} of a metric space ( X , d ) {\displaystyle (X,d)} , the following two statements are equivalent: The above follows directly from Jean Dieudonné , theorem 3.16.1, which states: For a metric space ( X , d ) {\displaystyle (X,d)} , 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

858-455: 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 ) {\displaystyle (X,d)} has

891-429: The necessity to consider special classes of spaces where this proposition is true. These spaces are said to have the Heine–Borel property . A metric space ( X , d ) {\displaystyle (X,d)} is said to have the Heine–Borel property if each closed bounded set in X {\displaystyle X} is compact. Many metric spaces fail to have the Heine–Borel property, such as

924-995: The power set P ⁡ ( S ) {\displaystyle \operatorname {\mathcal {P}} (S)} of a set S , the inclusion partial order is—up to an order isomorphism —the Cartesian product of k = | S | {\displaystyle k=|S|} (the cardinality of S ) copies of the partial order on { 0 , 1 } {\displaystyle \{0,1\}} for which 0 < 1. {\displaystyle 0<1.} This can be illustrated by enumerating S = { s 1 , s 2 , … , s k } , {\displaystyle S=\left\{s_{1},s_{2},\ldots ,s_{k}\right\},} , and associating with each subset T ⊆ S {\displaystyle T\subseteq S} (i.e., each element of 2 S {\displaystyle 2^{S}} )

957-736: The same meaning as and instead of the symbols ⊊ {\displaystyle \subsetneq } and ⊋ . {\displaystyle \supsetneq .} This usage makes ⊆ {\displaystyle \subseteq } and ⊂ {\displaystyle \subset } analogous to the inequality symbols ≤ {\displaystyle \leq } and < . {\displaystyle <.} For example, if x ≤ y , {\displaystyle x\leq y,} then x may or may not equal y , but if x < y , {\displaystyle x<y,} then x definitely does not equal y , and

990-501: The side length of T k is (2 a ) / 2 , 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  ∈  T k for each k . Since C covers T 0 , then it has some member U  ∈ C such that L  ∈ U . Since U

1023-423: The space H ( Ω ) {\displaystyle H(\Omega )} of holomorphic functions on an open set Ω ⊂ C n {\displaystyle \Omega \subset \mathbb {C} ^{n}} . More generally, any quasi-complete nuclear space has the Heine–Borel property. All Montel spaces have the Heine–Borel property as well. Subset In mathematics,

Heine–Borel theorem - Misplaced Pages Continue

1056-411: Was restricted to countable covers. Pierre Cousin (1895), Lebesgue (1898) and Schoenflies (1900) generalized it to arbitrary covers. If a set is compact, then it must be closed. Let S be a subset of R . 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 V U of

1089-435: 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 , 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

#670329