Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 5
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It is a closed setwhere each point is an accumulation point. Onthe other hand, it can be shown that the Cantorset can be placed in one-to-one correspondencewith the points of the interval [0, 1].(2) Any topological space homeomorphic tothe standard Cantor set in R1 .Cantor’s TheoremIf S is any set, there isno surjection from S onto the power set P(S).Cartan formulaA formula expressing therelationship between values of an operation ona product of terms and products of operations applied to individual terms. For the mod 2 Steenrod algebra, the Cartan formula is given bySq i (xy) =(Sq j x)(Sq i−j y).jThe sum is finite since Sq j x = 0 when j isgreater than the degree of the cohomology classx.
A differential in a spectral sequence is another example where there is a Cartan formula(if there is a product on the spectral sequence).Cartesian productFor any two sets X andY , the set, denoted X × Y , of all ordered pairs(x, y) with x ∈ X, y ∈ Y .Cartesian spaceThe standard coordinatespace Rn , where points are given by n realvalued coordinates for some n. Distance between two points x = (x1 , . .
. , xn ) and y =(y1 , . . . , yn ) is determined by the Pythagoreanidentity: nd(x, y) = (xi − yi )2 .i=1Cartesian space is a model of Euclidean geometry.catastrophe theoryThe study of quantitieswhich may change suddenly (discontinuously)even while the quantities that determine themchange smoothly.categorical theoryExample: When forces on an object grow tothe point of overcoming the opposing force dueto friction, the object will move suddenly.categorical theoryA consistent theory Tin a language L is categorical if all models ofT are isomorphic.
Because of the LöwenheimSkolem Theorem, no theory with an infinitemodel can be categorical in this sense, sincemodels of different cardinalities cannot be isomorphic.More generally, a consistent theory T is κcategorical for a cardinal κ if any two models ofT of size κ are isomorphic.category A category X consists of a class ofobjects, Obj(X), pairwise disjoint sets of functions (morphisms), HomX (A, B), for every ordered pair of objects A, B ∈Obj(X), and compositionsHomX (A, B)×HomX (B, C) → HomX (A, C) ,denoted (f, g) → gf satisfying the followingproperties:(i.) for each A ∈Obj(X) there is an identitymorphism 1A ∈ HomC (A, A) such that f 1A =f for all f ∈ HomX (A, B) and 1A g = g for allg ∈ HomX (C, A);(ii) associativity of composition for morphisms holds whenever possible: if f ∈HomX (A, B), g ∈ HomX (B, C), h ∈HomX (C, D), then h(gf ) = (hg)f .category of groupsThe class of all groupsG, H, .
. . , with each Hom(G, H ) equal to theset of all group homomorphisms f : G → H ,under the usual composition. Denoted Grp. Seecategory.category of linear spacesThe class of allvector spaces V , W, . . . , with each Hom(V , W )equal to the set of all linear transformations f :V → W , under the usual composition. DenotedLin. See category.category of manifolds The class of all differentiable manifolds M, N, .
. . , with eachHom(M, N ) equal to the set of all differentiablefunctions f : M → N , under the usual composition. Denoted Man. See category.© 2001 by CRC Press LLCcategory of ringsThe class of all ringsR, S, . . . , with each Hom(R, S) equal to the setof all ring homomorphisms f : R → S, under the usual composition. Denoted Ring. Seecategory.category of sets The class of all sets X, Y, . .
. ,with Hom(X, Y ) equal to the set of all functionsf : X → Y , under the usual composition. Denoted Set. See category.category of topological spacesThe classof all topological spaces X, Y, . . . , with eachHom(X, Y ) equal to the set of all continuousfunctions f : X → Y , under the usual composition. Denoted Top. See category.Cauchy sequence An infinite sequence {xn }of points in a metric space M, with distancefunction d, such that, given any positive number , there is an integer N such that for anypair of integers m, n greater than N the distanced(xm , xn ) is always less than .
Any convergentsequence is automatically a Cauchy sequence.Cavalieri’s TheoremThe theorem or principle that if two solids have equal area crosssections, then they have equal volumes, was published by Bonaventura Cavalieri in 1635. As aconsequence of this theorem, the volume of acylinder, even if it is oblique, is determined onlyby the height of the cylinder and the area of itsbase.cell A set whose interior is homeomorphic tothe n-dimensional unit disk {x ∈ Rn : x <1} and whose boundary is divided into finitelymany lower-dimensional cells, called faces ofthe original cell.
The number n is the dimensionof the cell and the cell itself is called an n-cell.Cells are the building blocks of a complex.central symmetryThe property of a geometric figure F , such that F contains a point c(the center of F ) so that, for every point p1 onF , there is another point p2 on F such that cbisects the line segment p1 p2 .centroid The point of intersection of the threemedians of a triangle.characteristic numberchainA formal finite linear combination ofsimplices in a simplicial complex K with integercoefficients, or more generally with coefficientsin some ring. The term is also used in moregeneral settings to denote an element of a chaincomplex.chain complex Let R be a ring (for example,the integers).
A chain complex of R-modulesconsists of a family of R-modules Cn , wheren ranges over the integers (or sometimes thenon-negative integers), together with homomorphisms ∂n : Cn −→ Cn−1 satisfying the condition: ∂n−1 ◦ ∂n (x) = 0 for every x in Cn .chain equivalent complexesLet C = {Cn }and C = {Cn } be chain complexes with boundary maps ∂ and ∂ , respectively. (See chaincomplex.) A chain mapping f : C −→ C is a chain equivalence if there is a chain mapping g : C −→ C and chain homotopies fromg ◦ f to the identity mapping of C and fromf ◦ g to the identity mapping of C . In this casewe say that C and C are chain equivalent.
Achain equivalence induces an isomorphism between the homology of C and the homology ofC . For example, if φ : X −→ Y is a homotopy equivalence of topological spaces, then φinduces a chain equivalence of the singular chaincomplexes of X and Y .chain group Let K be a simplicial complex.Then the nth chain group Cn (K) is the freeAbelian group constructed by taking all finitelinear combinations with integer coefficients ofn-dimensional simplices of K. Similarly, if Xis a topological space, the nth singular chaingroup is the free Abelian group constructed bytaking all finite linear combinations of singularsimplices, which are continuous functions fromthe standard n-dimensional simplex to X.chain homotopyLet C = {Cn } and C ={Cn } be chain complexes with boundary maps∂n and ∂n , respectively. Let f and g be chainmappings from C to C .
See chain complex,chain mapping. Then a chain homotopy T fromf to g is a collection of homomorphisms Tn :Cn −→ Cn+1such that ∂n+1 ◦ Tn + Tn−1 ◦ ∂n =fn − gn . For example, a homotopy between twomaps from a topological space X to a topologi-© 2001 by CRC Press LLCcal space Y induces a chain homotopy betweenthe induced chain maps from the singular chaincomplex of X to the singular chain complex ofY.chain mappingLet C = {Cn } and C ={Cn } be chain complexes with boundary maps∂n : Cn −→ Cn−1 and ∂n : Cn −→ Cn−1,respectively. See chain complex. A chain mapping f : C −→ C is a family of homomorphisms fn : Cn −→ Cn satisfying ∂n ◦ fn =fn−1 ◦ ∂n .
For example, when φ : X −→ Y iscontinuous, the induced map from the singularchain complex of X to the singular chain complex of Y is a chain map.characteristic class Let E −→ B be a vectorbundle. A characteristic class assigns a class ξin the cohomology H ∗ (B) of B to each vectorbundle over B so that the assignment is “predictable” or natural with respect to maps of vector bundles. That is, if the maps f : E −→ E and g : B −→ B form a map of vector bundlesso that E −→ B is equivalent to the pullbackg ∗ (E ) −→ B, then the class assigned to E −→B is the image of the class assigned to E −→ B under the map g ∗ : H ∗ (B ) −→ H ∗ (B).When the cohomology of the base space canbe considered as a set of numbers, the characteristic class is sometimes called a characteristicnumber.Example: Stiefel-Whitney classes of a manifold are characteristic classes in mod 2 cohomology.characteristic functionThe characteristicfunction χA of a set A of natural numbers is thefunction that indicates membership in that set;i.e., for all natural numbers n,1 if n ∈ AχA (n) =0 if n ∈ A.More generally, if A is a fixed universal setand B ⊆ A, then for all x ∈ A,1 if x ∈ BχB (x) =0 if x ∈ B.characteristic numberclass.See characteristicchoice functionchoice functionSuppose that {Xα }α∈ is afamily of non-empty sets.
Achoice function isa function f : {Xα }α∈ → α∈ Xα such thatf (Xα ) ∈ Xα for all α ∈ . See also Axiom ofChoice.choice set Suppose that {Xα }α∈ is a familyof pairwise disjoint, non-empty sets. A choiceset is a set Y , which consists of exactly one element from each set in the family.
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