Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 4
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(2) The binomial coefficient nk also represents the number of ways to choose k distinctitems from among n distinct items, without regard to the order of choosing. (3)The binomial coefficient nk is the kth entry in the nth row of Pascal’s Triangle. It must benoted that Pascal’s Triangle begins with row 0,and each row begins with entry 0. See Pascal’striangle.Binomial TheoremIf a and b are elementsof a commutative ring andin n is a non-negativeteger, then (a + b)n = nk=0 nk a k bn−k , wherenk is the binomial coefficient. See binomial coefficient.Bockstein operation In cohomology theory,a cohomology operation is a natural transformation between two cohomology functors. If0 → A → B → C → 0 is a short exact sequence of modules over a ring R, and if X ⊂ Yare topological spaces, then there is a long exactsequence in cohomology:· · · → H q (X, Y ; A) → H q (X, Y ; B) →H q (X, Y ; C) →H q+1 (X, Y ; A) → H q+1 (X, Y ; B) → .
. . .The homomorphismβ : H q (X, Y ; C) → H q+1 (X, Y ; A)is the Bockstein (cohomology) operation.bounded quantifierBolzano-Weierstrass TheoremEverybounded sequence in R has a convergent subsequence. That is, if{xn : n ∈ N} ⊆ [a, b]is an infinite sequence, then there is an increasing sequence {nk : k ∈ N} ⊆ N such that{xnk : k ∈ N} converges.Boolean algebraA non-empty set X, alongwith two binary operations ∪ and ∩ (called unionand intersection, respectively), a unary operation (called complement), and two elements0, 1 ∈ X which satisfy the following propertiesfor all A, B, C ∈ X.(i.) A ∪ (B ∪ C) = (A ∪ B) ∪ C(ii.) A ∩ (B ∩ C) = (A ∩ B) ∩ C(iii.) A ∪ B = B ∪ A(iv.) A ∩ B = B ∩ A(v.) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)(vi.) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)(vii.) A ∪ 0 = A and A ∩ 1 = A(viii.) There exists an element A so that A ∪A = 1 and A ∩ A = 0.Borel measurable functionA function f :X → Y , for X, Y topological spaces, such thatthe inverse image of any open set is a Borel set.This is equivalent to requiring the inverse imageof any Borel set to be Borel.
Any continuousfunction is Borel measurable.It is a theorem that f is Borel measurableif and only if f is a Baire function. See Bairefunction.Borel setThe collection B of Borel sets ofa topological space X is the smallest σ -algebracontaining all open sets of X. That is, in additionto containing open sets, B must be closed undercomplements and countable intersections (and,thus, is also closed under countable unions). Forcomparison, the topology on X is closed underarbitrary unions but only finite intersections.Borel sets may also be defined inductively:let 10 denote the collection of open sets and 01the closed sets. Then for 1 < α < ω1 , A ∈ α0if and only ifof B is in α0 . Then the collection of all Borelsets isB = ∪α<ω1 α0 = ∪α<ω1 0α .Sets in 20 are also known as Fσ sets; sets in 02are Gδ .If the space X is metrizable, then closed setsare Gδ and open sets are Fσ .
In this case, wehave for all α < ω1 ,0α0 ∪ 0α ⊆ α+1∩ 0α+1 .This puts the Borel sets in a hierarchy of lengthω1 known as the Borel hierarchy. See also projective set.bound(1) An upper bound on a set, S, ofreal numbers is a number u so that u ≥ s for alls ∈ S. If such a u exists, S is said to be boundedabove by u. Note that if u is an upper bound forthe set S, then so is any number larger than u.See also least upper bound.(2) A lower bound on a set, S, of real numbersis a number so that ≤ s for all s ∈ S. If suchan exists, S is said to be bounded below by .Note that if is a lower bound for the set S, thenso is any number smaller than . See greatestlower bound.(3) A bound on a set, S, of real numbers is anumber b so that |s| ≤ b for all s ∈ S.boundary group (homology)If {Cn , ∂n } isa chain complex (of Abelian groups), then thekth boundary group Bk is the subgroup of Ckconsisting of elements of the form ∂c for c inCk+1 .
That is, Bk = ∂Ck+1 .A = ∪n∈N Anboundary operator A chain complex {Cn , ∂n }consists of a sequence of groups or modulesover a ring R, together with homomorphisms∂n : Cn −→ Cn−1 , such that ∂n−1 ◦ ∂n = 0.The homomorphisms ∂n are called the boundaryoperators. Specifically, if K is an ordered simplicial complex and Cn is the free Abelian groupgenerated by the n-dimensional simplices, thenthe boundary operator is defined by taking anyn-simplex σ to the alternating sum of its n − 1dimensional faces. This definition is then extended to a homomorphism.where, for each n ∈ N, An ∈ 0αn and αn < α.A set B is in 0α if and only if the complementbounded quantifier The quantifiers ∀x < yand ∃x < y.
The statement ∀x < y φ(x) is© 2001 by CRC Press LLCbound variableequivalent to ∀x(x < y → φ(x)), and ∃x <y φ(x) is equivalent to ∃x(x < y ∧ φ(x)).More generally, ∀x ∈ y φ(x) is equivalentto ∀x(x ∈ y → φ(x)) and ∃x ∈ y φ(x) isequivalent to ∃x(x ∈ y ∧ φ(x)).Intuitively, if a piece of paper is taken off atable, crumpled up, and laid back down on thesame part of the table, then at least one point isexactly above the same point on the table that itwas originally.bound variableLet L be a first-order language and let ϕ be a well-formed formula of L.An occurrence of a variable v in ϕ is bound ifit occurs as the variable of a quantifier or withinthe scope of a quantifier ∀v or ∃v.
The scope ofthe quantifier ∀v in a formula ∀vα is α.For example, the first occurrence of the variable v1 is free, while the remaining occurrencesare bound in the formulabundle groupA group that acts (continuously) on a vector bundle or fiber bundle E −→B and preserves fibers (so the action restricts toan action on each inverse image of a point in B).For example, the real orthogonal group O(n) isa bundle group for any rank n real vector bundle.If the bundle is orientable, then SO(n) is also abundle group for the vector bundle.The bundle group may also be called the structure group of the bundle.∀v2 (v1 = v2 → ∀v1 (v1 = v3 )).All occurrences of the variable v1 are bound inthe formula∀v1 (v1 = v2 → ∀v1 (v1 = v3 )).box topologyproductA topology on the CartesianXαα∈Aof a collection of topological spaces Xα , havingas a basis the set of all open boxes, α∈A Uα ,where each Uα is an open subset of Xα . The difference between this and the product topology isthat in the box topology, there are no restrictionson any of the Uα .Brouwer Fixed-Point TheoremAny continuous mapping f of a finite product of copiesof [0, 1] to itself, or of S n to itself, has a fixedpoint, that is, a point z such that f (z) = z.© 2001 by CRC Press LLCbundle mapping A fiber preserving map g :E −→ E , where p : E −→ B and p : E −→B are fiber bundles.
If the bundles are smoothvector bundles, then g must be a smooth mapand linear on the vector space fibers.Example: When a manifold is embedded inRn , it has both a tangent and a normal bundle.The direct sum of these is the trivial bundle M ×Rn ; each inclusion into the trivial rank n bundleis a bundle mapping.bundle of planes A fiber bundle whose fibersare all homeomorphic to R2 . A canonical example of this is given by considering the Grassmann manifold of planes in Rn .
Each pointcorresponds to a plane in Rn in the same wayeach point of the projective space RPn−1 corresponds to a line in Rn . The bundle of planesover this manifold is given by allowing the fiberover each point in the manifold to be the actualplane represented by that point. If one considers the manifold as the collection of names ofthe planes, then the bundle is the collection ofplanes, parameterized by their “names”.catastrophe theoryCcanonical bundleIf the points of a spacerepresent (continuously parameterized) geometric objects, then the space has a canonical bundle given by setting the fiber above each pointto be the geometric object to which that pointcorresponds. Examples include the canonicalline bundle of projective space and the canonical vector bundle over a Grassmann manifold(the manifold of affine n-spaces in Rm ).canonical line bundle Projective space RPncan be considered as the space of all lines inRn+1 which go through the origin or, equivalently, as the quotient of S n+1 formed by identifying each point with its negative.
The canonical line bundle over RPn is the rank one vectorbundle formed by taking as fiber over a point inRPn the actual line that the point represents.Example: RP1 is homeomorphic to S 1 ; thecanonical line bundle over RP1 is homeomorphic to the Möbius band.There are also projective spaces formed overcomplex or quaternionic space, where a line isa complex or quaternionic line.Cantor-Bernstein TheoremIf A and B aresets, and f : A → B, g : B → A are injectivefunctions, then there exists a bijection h : A →B. This theorem is also known as the CantorSchröder-Bernstein Theorem or the SchröderBernstein Theorem.Cantor-Schröder-Bernstein TheoremCantor-Bernstein Theorem.SeeCantor set (1) (The standard Cantor set.) Asubset of R1 which is an example of a totally disconnected compact topological space in whichevery element is a limit point of the set.To construct the Cantor set as a subset of[0, 1], let I0 = [0, 1] ⊂ R1 , I1 = [0, 13 ] ∪ [ 23 , 1]and I2 = [0, 19 ]∪[ 29 , 13 ]∪[ 23 , 79 ]∪[ 89 , 1].
In general, define In to be the union of closed intervalsobtained by removing the open “middle thirds”© 2001 by CRC Press LLCfrom each of the closed intervals comprisingIn−1 . The Cantor set is defined as C = ∩∞n=1 In .The Cantor set has length 0, which can beverified by summing the lengths of the intervalsremoved to obtain a sum of 1.
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