Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 7
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For example, the sets [a, b] and{a} are closed in the usual topology of the realline.(2) A closed set of ordinals is one that isclosed in the order topology. That is, C ⊆ κis closed if, for any limit ordinal λ < κ, if C ∩ λis unbounded in λ, then λ ∈ C. Equivalently, if© 2001 by CRC Press LLC{βα : α < λ} ⊆ C is an increasing sequence oflength λ < κ, thenβ = lim βα ∈ C .α→λFor example, the set of all limit ordinals lessthan κ is closed in κ. See also unbounded set,stationary set.closed surfaceA compact Hausdorff topological space with the property that each pointhas a neighborhood topologically equivalent tothe plane.
Thus, a closed surface is a compact2-dimensional manifold without boundary. The222ellipsoids given by xa 2 + yb2 + cz2 − 1 = 0 aresimple examples of closed surfaces. More generally, if f (x, y, z) is a differentiable function,then the set of points S satisfying f (x, y, z) = 0is a closed surface provided that S is boundedand the gradient of f does not vanish at any pointin S.closure of a setThe closure of a subset Aof a topological space X is the smallest closedset Ā ⊆ X which contains A.
In other words,Ā is the intersection of all closed sets in X thatcontain A. Equivalently, Ā = A ∪ A , where Ais the derived set of A. For example, the closureof the rationals in the usual topology is the wholereal line.cluster pointSee accumulation point.cobordismA cobordism between two ndimensional manifolds is an (n+1)-dimensionalmanifold whose boundary is the disjoint unionof the two lower dimensional manifolds.
Acobordism between two manifolds with a certain structure must also have that structure. Forexample, if the manifolds are real oriented manifolds, then the cobordism must also be a realoriented manifold.Example: The cylinder provides a cobordismbetween the circle and itself. Any manifoldwith boundary provides a cobordism betweenthe boundary manifold and the empty set, whichis considered an n-manifold for all n.cobordism class For a manifold M, the classof all manifolds cobordant M, that is, all manifolds N for which there exists a manifold Wcomb spacewhose boundary is the disjoint union of M andN.cobordism group The cobordism classes ofn-dimensional manifolds (possibly with additional structure) form an Abelian group; the product is given by disjoint union.
The identity element is the class given by the empty set. Theinverse of the cobordism class of a manifold Mis given by reversing the orientation of M; themanifold M × [0, 1] is a cobordism between Mand M with the reverse orientation. (See cobordism class.) When studying cobordism classesof unoriented manifolds, each manifold is itsown inverse; thus, all such cobordism classesare 2-torsion.Some results in geometry show that cobordant manifolds may have a common geometricor topological property, for example, two spincobordant manifolds either both admit a positivescalar curvature metric, or neither manifold canhave such a metric.Codazzi-Mainardi equationsA system ofpartial differential equations arising in the theory of surfaces.
If M is a surface in R3 withlocal coordinates (u1 , u2 ), its geometric invariants can be described by its first fundamentalform gij (u1 , u2 ) and second fundamental formLij (u1 , u2 ). The Christoffel symbols ijk aredetermined by the first fundamental form. (SeeChristoffel symbols.) In order for functions gijand Lij , i, j = 1, 2 to be the first and secondfundamental forms of a surface, certain integrability conditions (arising from equality of mixedpartial derivatives) must be satisfied. One set ofconditions, the Codazzi-Mainardi equations, isgiven in terms of the Christoffel symbols by:∂Lij∂Likl−+ ikLlj − ijl Llk = 0 .j∂u∂ukcodimensionA nonnegative integer associated with a subspace W of a space V .
Wheneverthe space has a dimension (e.g., a topologicalor a vector space) denoted by dimV , the codimension of W is the defect dimV −dimW . Forexample, a curve in a surface has codimension 1(topology) and a line in space has codimension2 (a line through the origin is a vector subspaceR of R3 ).© 2001 by CRC Press LLCcofinalLet α, β be limit ordinals. An increasing sequence ατ : τ < β is cofinal in αif limτ →β ατ = α. See limit ordinal.cofinalityLet α be an infinite limit ordinal.The cofinality of α is the least ordinal β suchthat there exists a sequence ατ : τ < β whichis cofinal in α.
See cofinal.cofinite subsetA subset A of an infinite setS, such that S\A is finite. Thus, the set of all integers with absolute value at least 13 is a cofinitesubset of Z.coimageLet C be an additive category andf ∈ HomC (X, Y ) a morphism. If i ∈ HomC (X ,X) is a morphism such that f i = 0, then a coimage of f is a morphism g ∈ HomC (X, Y ) suchthat gi = 0. See additive category.coinfinite subset A subset A if an infinite setS such that S\A is infinite. Thus, the set of alleven integers is a coinfinite subset of Z.collapseA collapse of a complex K is a finite sequence of elementary combinatorial operations which preserves the homotopy type ofthe underlying space.For example, let K be a simplicial complexof dimension n of the form K = L ∪ σ ∪ τ ,where L is a subcomplex of K, σ is an openn-simplex of K, and τ is a free face of σ . Thatis, τ is an n − 1 dimensional face of σ and is notthe face of any other n-dimensional simplex.The operation of replacing the complex L ∪σ ∪ τ with the subcomplex L is called an elementary collapse of K and is denoted K L.A collapse is a finite sequence of elementarycollapses K L1 · · · Lm .When K is a CW complex, ball pairs of theform (B n , B n−1 ) are used in place of the pair(σ, τ ).collectionSee set.collinearPoints that lie on the same line oron planes that share a common line.comb spaceA topological subspace of theplane R2 which resembles a comb with infinitelymany teeth converging to one end.
For example,common tangentthe subset of the unit square [0, 1]2 given by({0} × [0, 1])∪1({ : k ≥ 1} × [0, 1]) ∪ ([0, 1] × {0})kis a comb space. The subspace obtained fromthis set by deleting the line segment {0} × (0, 1)is an example of a connected set that is not pathconnected.common tangenttwo circles.A line that is tangent tocommutative diagramf1Ag1 −→C−→f2A diagramBg2Din which the two compositions g2 f1 and f2 g1are equal.
Commutative triangles can be considered a special case if one of the functionsis the identity. Larger diagrams composed ofsquares and triangles commute if each squareand triangle inside the diagram commutes. Seediagram.compact(1) The property of a topologicalspace X that every cover of X by open sets (every collection {Xα } of open sets with X ⊂ ∪Xα )contains a finite subcover (a finite collectionXα1 , . . . , Xαn with X ⊂ ∪Xαi ).(2) A compact topological space.compact complex manifold A complex manifold which is compact in the complex topology.A common example is a Riemann surface (1dimensional complex manifold): the (Riemann)sphere is compact, unlike the sphere with a pointremoved.
The sphere with an open disk removedis also compact in the complex topology, butstrictly speaking it is not a complex manifold(some points do not lie in an open disk): it isknown as a manifold with boundary. See complex manifold.compactificationA compactification of atopological space X is a pair, (Y, f ), where Y isa compact Hausdorff space and f is a homeomorphism from X onto a dense subset of Y .
A© 2001 by CRC Press LLCnecessary and sufficient condition for a space tohave a compactification is that it be completelyregular. See also one-point compactification,Stone-Čech compactification.compact leafA concept arising in the theory of foliations. A foliated manifold is an ndimensional manifold M, partitioned into a family of disjoint, path-connected subsets Lα suchthat there is a covering of M by open sets Ui andhomeomorphisms hi : Ui −→ Rn taking eachcomponent of Lα ∩Ui onto a parallel translate ofthe subspace Rk . Each Lα is called a leaf, and itis a compact leaf if it is compact as a subspace.compact-open topology The topology on thespace of continuous functions from a topologicalspace X to a topological space Y , generated bytaking as a subbasis all sets of the form {f :f (C) ⊆ U }, where C ⊆ X is compact andU ⊆ Y is open. If Y is a metric space, thistopology is the same as that given by uniformconvergence on compact sets.comparability of cardinal numbersTheproposition that, for any two cardinals α, β, either α ≤ β or β ≤ α.compassAn instrument for constructingpoints at a certain distance from a fixed pointand for measuring distance between points.compatible (elements of a partial ordering)Two elements p and q of a partial order (P, ≤)such that there is an r ∈ P with r ≤ p andr ≤ q.
Otherwise p and q are incompatible.In the special case of a Boolean algebra, pand q are compatible if and only if p ∧ q = 0.In a tree, however, p and q are compatible if andonly if they are comparable: p ≤ q or q ≤ p.complementary angles Two angles are complementary if their sum is a right angle.complement of a set If X is a set contained ina universal set U , the complement of X, denotedX , is the set of all elements in U that do notbelong to X. More precisely, X = {u ∈ U :u∈/ X}.complex conjugate bundlecompletely additive function An arithmeticfunction f having the property that f (mn) =f (m) + f (n) for all positive integers m andn.
(See arithmetic function.) For example, thefunction f (n) = log n is completely additive.The values of a completely additive function depend only on its values at primes, since f (pi ) =i · f (p). See also additive function.the usual set {¬, ∧, ∨, →, ↔} of logical connectives, there is a well-formed propositionalformula ψ, whose logical connectives are fromC, such that ϕ and ψ are logically equivalent.Examples of complete sets of logical connectives include {¬, ∧, ∨}, {¬, ∧}, {¬, ∨}, and{¬, →}.
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