Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 12
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Assume there exists an open covering {Uα } (α ∈ ) of B, anda homeomorphism φα : Uα XF ≈ p−1 (Uα ) foreach α ∈ . Define φα,β : F ≈ p −1 (b) (b ∈Uα ) by φα,β (y) = φα (b, y). Then gβα (b) =−1φβ,bφα,b ∈ G for b ∈ Uα ∩Uβ . Then the continuous transformation gβα is the coordinate transformation of the coordinate bundle (E, p, B, F,G, Uα , φα ) belonging to the fiber bundle(E, p, B, F, G).coplanarLying in the same plane.coproductFor any two sets X and Y , a coproduct of X and Y is a disjoint union D in thediagramijX −→ D ←− Y,where i and j are injections.
The set D is notunique and can be constructed as follows. If Xand Y are disjoint, then let D = X ∪ Y . If X andY are not disjoint, let D = X ∪ Y , where Y isa set that is equivalent to Y and disjoint from X.Other coproducts of X and Y can be formed bychoosing different sets Y .corresponding angles Let two straight lineslying in R2 be cut by a transversal, so that anglesx and y are a pair of alternating interior angles,and y and z are a pair of vertical angles.
Then xand z are corresponding angles.cotangent bundle Let M be an n-dimensionaldifferentiable manifold of class C r . ConsiderT (M), the union over p ∈ M of all the vectorspaces Tp (M) of vectors tangent to M at p. Define π : T (M) → M by π(Tp (M)) = p. ThenT (M) may be regarded as a manifold called thetangent bundle of M. The dual bundle is thecotangent bundle.coterminal angles Angles with the same terminal side (the moving straight line which revolves around the fixed straight line, the initialside, to form the angle) and the same initial side.Two angles are coterminal if they are generatedby the revolution of two lines about the samepoint of the initial side in such a way that the final positions of the revolving lines are the same.For example, 60◦ and -300◦ are coterminal angles.© 2001 by CRC Press LLCCountable Axiom of Choice The statementthat, for any countable family of non-empty,pairwise disjoint sets {Xα }α∈ , there exists a setY which consists of exactly one element fromeach set in the family.
Equivalently, if {Xα }α∈is a countable family of non-empty sets,thenthere exists a function f : {Xα }α∈ → α∈Xα such that f (Xα ) ∈ Xα for all α ∈ . SeeAxiom of Choice.countable chain condition(1) A partial order (P, ≤) has the countable chain condition ifany antichain in P is countable. A set A ⊆ P isan antichain if its elements are pairwise incompatible; that is, for any p and q in A, there is nor ∈ P with r ≤ p and r ≤ q. Thus, P has thecountable chain condition (or is ccc) if it has nouncountable subset of pairwise incompatible elements.
Examples of ccc partial orders includethe collection of all finite sequences of 0s and 1sordered by extension (p ≤ q if p ⊇ q), and thecollection of all measurable sets modulo measure zero sets ordered by inclusion ([A] ≤ [B]if A ⊆ B).(2) A topological space satisfies the countable chain condition if it contains no uncountable collection of pairwise disjoint non-emptyopen sets. If X is a topological space and P isthe collection of all non-empty open subsets ofX ordered by p ≤ q if and only if p ⊆ q, thenX is a ccc topological space if and only if P isa ccc partial order.countably compact topological spaceAtopological space X such that any countable opencover of X contains a finite subcover. That is,if each Un is open and X = ∪n∈N Un , then thereis a finite set A ⊆ N with X = ∪n∈A Un . Thisis equivalent to requiring that any countably infinite subset of X has an accumulation point.The space of ordinals less than ω1 with theorder topology is a countably compact spacewhich is not compact.counterclockwiseThe direction of rotationopposite to that in which the hands of the clockmove.covariant functor Let C and D be categories.A covariant functor F is a function F : Obj(C)covering dimension→ Obj(D) such that, for any A, B ∈Obj(C) thefollowing hold:(i.) if f ∈ HomC (A, B), thenF (f ) ∈ HomD (F (A), F (B)) ;(ii.) F (1C ) = 1D ;(iii.) F (gf ) = F (g)F (f ), where f and gare morphisms in C whose composition gf isdefined.covering dimensionA nonnegative integer,assigned to a set by means of coverings.
Fortopological spaces the covering dimension (orLebesque dimension) is defined in terms of opencoverings. The dimension of a normal space xis less than or equal to n if, in each finite opencovering of x, a finite open covering can be inscribed whose number of elements containing agiven point is less than or equal to n + 1.covering groupp : E −→ X is a coveringspace of X if every point of X has a neighborhood whose inverse image is the disjoint unionof open sets homeomorphic to the neighborhoodby p. The covering group (group of covering transformations) is the group of homeomorphisms of E which preserve the fibers (homeomorphisms h : E −→ E with ph = p).The real line covers the circle S 1 by the mapwhich takes x to e2πix .
The group of coveringtransformations is isomorphic to the group ofintegers.covering homotopy propertyA map p : E−→ B has the covering homotopy property if itsatisfies the following: given any map f : X −→E and any homotopy h : X × [0, 1] −→ B ofp ◦ f (so hi = pf ), there is a lift H : X ×[0, 1] −→ E of h so that pH = h and H i = f ,where i is the inclusion of X × {0} in X × [0, 1].That is, given maps represented by the solidlines in the diagram below,fXi↓−→X×I−→hE↓pBthere exists a map H : X × I −→ E so that thediagram with H added commutes.A surjective map satisfying the covering homotopy property is called a fibration. This is© 2001 by CRC Press LLCa generalization of the concept of fiber bundle;the fiber in a fibration is only determined up tohomotopy, that is, the inverse images of two different points (if B is connected) will only behomotopy equivalent to each other.covering mapA continuous surjection p :X → Y such that every y ∈ Y has an openpath connected neighborhood U such that, foreach open path component V ⊆ p−1 (U ), p isa homeomorphism from V onto U .
In otherwords, the V s form a stack of copies over Uwhich cover it. For example, the map p(t) =(cos t, sin t) is a covering map from R to the unitcircle.covering spaceA topological space X is acovering space of Y if there is a covering mapp : X → Y . See covering map. For example, Ris a covering space of the unit circle via p(t) =(cos t, sin t).covering transformation A map φ : E −→E such that p ◦ φ = φ, where p : E −→ B is acovering map of an arcwise connected, locallyarcwise connected space B. See covering map.covering transformation groupThe groupof covering transformations φ : E −→ E, under composition.
See covering transformation.In the special case where E is the universal coverof B, this group is isomorphic to the fundamental group of the space B.cross-sectionA cross-section or section ofa fiber bundle p : E −→ B with fiber F is amap s : B −→ E with ph equal to the identityon B. Clearly every trivial bundle B × F hasnumerous sections. A non-trivial example is theMöbius band, a bundle over its middle circle.The inclusion of the middle circle is a sectionfor that bundle.cubeOne of the five types of regular polyhedra in E 3 . Also known as a hexahedron, itis a solid bounded by 6 planes with 12 equaledges and face angles that are right angles. InE n , it is a set consisting of all those points x =(x1 , ..., xn ), for which xi is such that ai ≤ xi ≤bi for each i, where ai and bi are such that bi −aihas the same value for each i.cycle groupcumulative hierarchy The hierarchy of setsdefined recursively using the power set operation at successor stages and union at limit stages:(i.) V0 = Ø, (ii.) Vα+1 = P(Vα ), for all ordinals α, and (iii.) Vλ = β<λ Vβ , for any limitordinal λ.
Also known as the Zermelo hierarchy.See also universe of sets.curvature A measure of the quantitative characteristics (in terms of numbers, vectors, tensors) which describe the degree to which someobject (a curve, Riemannian manifold, etc.) deviates from certain other objects (a straight line,a Euclidean surface, etc.), which are considered to be flat. As a local property of a planecurve, curvature may intuitively be thought ofas the degree to which a curve is “bent” at eachpoint. For non-planar space curves, curvatureis defined as the magnitude of a rate of rotation vector.
Gauss had defined the curvature ofa surface in R3 at a point (x, y, z) as the limitof the ratio of the area of the region on a unitsphere around a point (X, Y, Z) [determined bythe radius of the sphere in a direction normal to(x, y, z)] to the area of the region on the surface around (x, y, z), as these two areas shrinkto their respective points. Riemann’s conception of curvature for any n-dimensional manifold was a generalization of Gauss’ definitionfor surfaces.curve A continuous function from (an interval in) R into Rn , although usually referred toas the image (range) of such a function.Euclid distinguished lines from curves, buttoday lines, in the Euclidean sense, are considered curves which include straight lines. GeorgCantor defined a curve as a continuum that isnowhere dense in R2 . A continuous curve in R2that covers a square is a Peano curve.curve of constant breadthLet E be theboundary of a convex body X in R2 , O an interior point of X, and P an arbitrary point ofX different from O.
E admits exactly one supporting line l(P ) which is perpendicular to theline OP and meets the half-line OP . Let OPbe the half-line with direction opposite to that ofOP , and l the supporting line l(P ). Then thedistance between the parallel lines l and l is thebreadth of E. Let M = M(E) be the maximum© 2001 by CRC Press LLCand m = m(E) be the minimum of the breathof E. If M = m, then E is a curve of constantbreadth.cuspA double point on a curve C, at whichtwo tangents to C are coincident.cut point (1) In topology, a point p in a spaceX such that X\{p} = A ∪ B, where A and Bare nonempty open sets.(2) In Riemannian geometry, if M is a Riemannian manifold and p is a point of M, thena point q of M is a cut point with respect top if there is a shortest geodesic joining p to qwhich, if extended beyond q, fails to be a shortest path to points beyond q.
For example, on thestandard sphere the antipode of any point is theunique cut point.CW complex A topological space constructed iteratively as follows. Let D n be an n-cell,that is, a set homeomorphic to all points of distance at most 1 from the origin in Rn ; the boundary of D n is the (n−1)-sphere, S n−1 . Begin witha collection of points. At each stage, attach newcells by identifying the boundary of a cell D nwith points in lower dimensional cells.One can build the sphere S n in two distinctways.
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