Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 2
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The contravariant functor Hom∗ : X → Sets maps A ∈Obj(X )to HomX (A, X); f ∈ HomX (A, A ) is mappedtof ∗ : HomX (A , X) → HomX (A, X) ,by g → gf .Let C, D be categories. Two covariant functors F : C → D and G : D → C are adjointfunctors if, for any A, A ∈ Obj(C), B, B ∈Obj(D), there exists a bijectionφ : HomC (A, G(B)) → HomD (F (A), B)that makes the following diagrams commute forany f : A → A in C, g : B → B in D:algebraic varietyHomC (A, G(B))φHomD (F (A), B)HomC (A, G(B))φHomD (F (A), B)f∗−→(F (f ))∗HomC (A , G(B))φ−→HomD (F (A ), B)(G(g))∗−→HomC (A, G(B ))φg∗HomD (F (A), B )−→See category of sets.alephs Form the sequence of infinite cardinalnumbers (ℵα ), where α is an ordinal number.Alexander’s Horned Sphere An example ofa two sphere in R3 whose complement in R3 isnot topologically equivalent to the complementof the standard two sphere S 2 ⊂ R3 .This space may be constructed as follows:On the standard two sphere S 2 , choose two mutually disjoint disks and extend each to form two“horns” whose tips form a pair of parallel disks.On each of the parallel disks, form a pair ofhorns with parallel disk tips in which each pairof horns interlocks the other and where the distance between each pair of horn tips is half theprevious distance.
Continuing this process, atstage n, 2n pairwise linked horns are created.In the limit, as the number of stages of theconstruction approaches infinity, the tips of thehorns form a set of limit points in R3 homeomorphic to the Cantor set. The resulting surface ishomeomorphic to the standard two sphere S 2 butthe complement in R3 is not simply connected.algebra of sets A collection of subsets S of anon-empty set X which contains X and is closedwith respect to the formation of finite unions,intersections, and differences. More precisely,(i.) X ∈ S;(ii.) if A, B ∈ S, then A ∪ B, A ∩ B, andA\B are also in S.See union, difference of sets.algebraic number(1) A complex numberwhich is a zero of a polynomial with rational coefficients (i.e., α is algebraic if there exist ratio-© 2001 by CRC Press LLCAlexander’s Horned Sphere.PovRay.Graphic rendered bynnal numbers a0 , a1 , .
. . , an so thatai α i = 0).i=0√For example, 2 is an algebraic number sinceit satisfies the equation x 2 − 2 = 0. Since thereis no polynomial p(x) with rational coefficientssuch that p(π ) = 0, we see that π is not an algebraic number. A complex number that is notan algebraic number is called a transcendentalnumber.(2) If F is a field, then α is said to be algebraic over F if α is a zero of a polynomialhaving coefficients in F . That is, if there existelements f0 , f1 , f2 , .
. . , fn of F so that f0 +f1 α + f2 α 2 · · · + fn α n = 0, then α is algebraicover F .algebraic number fieldA subfield of thecomplex numbers consisting entirely of algebraic numbers. See also algebraic number.algebraic number theoryThat branch ofmathematics involving the study of algebraicnumbers and their generalizations. It can be argued that the genesis of algebraic number theorywas Fermat’s Last Theorem since much of theresults and techniques of the subject sprung directly or indirectly from attempts to prove theFermat conjecture.algebraic variety Let A be a polynomial ringk[x1 , . . .
, xn ] over a field k. An affine algebraicvariety is a closed subset of An (in the Zariskitopology of An ) which is not the union of twoproper (Zariski) closed subsets of An . In theZariski topology, a closed set is the set of common zeros of a set of polynomials. Thus, anaffine algebraic variety is a subset of An whichis the set of common zeros of a set of polynomi-altitudeals but which cannot be expressed as the unionof two such sets.The topology on an affine variety is inheritedfrom An .In general, an (abstract) algebraic variety is atopological space with open sets Ui whose unionis the whole space and each of which has anaffine algebraic variety structure so that the induced variety structures (from Ui and Uj ) oneach intersection Ui ∩ Uj are isomorphic.The solutions to any polynomial equation forman algebraic variety.
Real and complex projective spaces can be described as algebraic varieties (k is the field of real or complex numbers,respectively).altitudeIn plane geometry, a line segmentjoining a vertex of a triangle to the line throughthe opposite side and perpendicular to the line.The term is also used to describe the length ofthe line segment. The area of a triangle is givenby one half the product of the length of any sideand the length of the corresponding altitude.amicable pair of integersTwo positive integers m and n such that the sum of the positivedivisors of both m and n is equal to the sum ofm and n, i.e., σ (m) = σ (n) = m + n. Forexample, 220 and 284 form an amicable pair,sinceσ (220) = σ (284) = 504 .A perfect number forms an amicable pair withitself.analytic number theory That branch of mathematics in which the methods and ideas of realand complex analysis are applied to problemsconcerning integers.analytic set The continuous image of a Borelset.
More precisely, if X is a Polish space andA ⊆ X, then A is analytic if there is a Borel set Bcontained in a Polish space Y and a continuousf : X → Y with f (A) = B. Equivalently, Ais analytic if it is the projection in X of a closedsetC ⊆ X × NN ,where NN is the Baire space. Every Borel set isanalytic, but there are analytic sets that are not© 2001 by CRC Press LLCBorel. The collection of analytic sets is denoted11 . See also Borel set, projective set.annulus A topological space homeomorphicto the product of the sphere S n and the closedunit interval I .
The term sometimes refers specifically to a closed subset of the plane bounded bytwo concentric circles.antichainA subset A of a partially orderedset (P , ≤) such that any two distinct elementsx, y ∈ A are not comparable under the ordering≤. Symbolically, neither x ≤ y nor y ≤ x forany x, y ∈ A.arcA subset of a topological space, homeomorphic to the closed unit interval [0, 1].arcwise connected component If p is a pointin a topological space X, then the arcwise connected component of p in X is the set of pointsq in X such that there is an arc (in X) joiningp to q. That is, for any point q distinct fromp in the arc component of p there is a homeomorphism φ : [0, 1] −→ J of the unit intervalonto some subspace J containing p and q. Thearcwise connected component of p is the largestarcwise connected subspace of X containing p.arcwise connected topological space A topological space X such that, given any two distinctpoints p and q in X, there is a subspace J of Xhomeomorphic to the unit interval [0, 1] containing both p and q.arithmetical hierarchy A method of classifying the complexity of a set of natural numbersbased on the quantifier complexity of its definition.
The arithmetical hierarchy consists ofclasses of sets n0 , 0n , and 0n , for n ≥ 0.A set A is in 00 = 00 if it is recursive (computable). For n ≥ 1, a set A is in n0 if there isa computable (recursive) (n + 1)–ary relation Rsuch that for all natural numbers x,x ∈ A ⇐⇒ (∃y1 )(∀y2 ) . . . (Qn yn )R(x, y),where Qn is ∃ if n is odd and Qn is ∀ if n isodd, and where y abbreviates y1 , . . . , yn . Forn ≥ 1, a set A is in 0n if there is a computable(recursive) (n + 1)–ary relation R such that foratom of a Boolean algebraall natural numbers x,x ∈ A ⇐⇒ (∀y1 )(∃y2 ) .
. . (Qn yn )R(x, y),where Qn is ∃ if n is even and Qn is ∀ if n isodd. For n ≥ 0, a set A is in 0n if it is in bothn0 and 0n .Note that it suffices to define the classes n0and 0n as above since, using a computable coding function, pairs of like quantifiers (for example, (∃y1 )(∃y2 )) can be contracted to a singlequantifier ((∃y)). The superscript 0 in n0 , 0n ,0n is sometimes omitted and indicates classesin the arithmetical hierarchy, as opposed to theanalytical hierarchy.A set A is arithmetical if it belongs to thearithmetical hierarchy; i.e., if, for some n, Ais in n0 or 0n .
For example, any computably(recursively) enumerable set is in 10 .arithmetical setA set A which belongs tothe arithmetical hierarchy; i.e., for some n, Ais in n0 or 0n . See arithmetical hierarchy. Forexample, any computably (recursively) enumerable set is in 10 .arithmetic functionA function whose domain is the set of positive integers. Usually, anarithmetic function measures some property ofan integer, e.g., the Euler phi function φ or thesum of divisors function σ .
The properties ofthe function itself, such as its order of growth orwhether or not it is multiplicative, are often studied. Arithmetic functions are also called numbertheoretic functions.Aronszajn treeA tree of height ω1 whichhas no uncountable branches or levels. Thus,for each α < ω1 , the α-level of T , Levα (T ),given byt ∈ T : ordertype({s ∈ T : s < t}) = αis countable, Levω1 (T ) is the first empty level ofT , and any set B ⊆ T which is totally orderedby < (branch) is countable.
An Aronszajn treeis constructible in ZFC without any extra settheoretic hypotheses.For any regular cardinal κ, a κ-Aronszajn treeis a tree of height κ in which all levels have sizeless than κ and all branches have length less thanκ. See also Suslin tree, Kurepa tree.© 2001 by CRC Press LLCassociated fiber bundleA concept in thetheory of fiber bundles. A fiber bundle ζ consists of a space B called the base space, a spaceE called the total space, a space F called thefiber, a topological group G of transformationsof F , and a map π : E −→ B.
There is acovering of B by open sets Ui and homeomorphisms φi : Ui × F −→ Ei = π −1 (Ui ) suchthat π ◦ φi (x, V ) = x. This identifies π −1 (x)with the fiber F . When two sets Ui and Uj overlap, the two identifications are related by coordinate transformations gij (x) of F , which arerequired to be continuously varying elements ofG. If G also acts as a group of transformationson a space F , then the associated fiber bundleζ = π : E −→ B is the (uniquely determined) fiber bundle with the same base spaceB, fiber F , and the same coordinate transformations as ζ .associated principal fiber bundle The associated fiber bundle, of a fiber bundle ζ , with thefiber F replaced by the group G. See associatedfiber bundle.
The group acts by left multiplication, and the coordinate transformations gij arethe same as those of the bundle ζ .atomic formulaLet L be a first order language. An atomic formula is an expressionwhich has the form P (t1 , . . . , tn ), where P isan n-place predicate symbol of L and t1 , . . . , tnare terms of L.
If L contains equality (=), then= is viewed as a two-place predicate. Consequently, if t1 and t2 are terms, then t1 = t2 is anatomic formula.atomic modelA model A in a language Lsuch that every n-tuple of elements of A satisfies a complete formula in T , the theory ofA. That is, for any ā ∈ An , there is an Lformula θ (x̄) such that A |= θ (ā), and for anyL-formula φ, either T ∀ x̄ θ (x̄) → φ(x̄) orT ∀x̄ θ (x̄) → ¬φ(x̄) .
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