Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 36
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The Steenrod square andpower operations are examples of stable (primary) cohomology operations. See Steenrodsquare operation.for which(i.) Sq 0 is the identity;(ii.) for u in H i (X, A), Sq i u = u ∪ u = u2 ;(iii.) for u in H i (X, A) and i > k, Sq k u = 0;(iv.) for u in H a (X, A) and v in H b (Y, B),the effect on uv in H ∗ (X × Y, A × B) is givenby the Cartan formula:Sq k (u × v) =Sq i u × Sq j v .stable rangeSome algebraic invariants behave well with respect to suspension, sometimeswith connectivity restrictions.
For example, ifX is (n − 1)-connected and i ≤ 2n − 2, thenπi (X) is isomorphic to πi+1 (X). This rangeis called the stable range of X.In homotopy theory, one may be concernedwith the stable range in calculating homotopygroups or the effect of cohomology operations.i+j =kstable secondary cohomology operationstable (primary) cohomology operation.Seestably parallelizable manifoldA smoothmanifold M such that the Whitney sum of thetangent bundle of M and a trivial bundle over Mis a trivial bundle. For example, the tangent bundle of the sphere S 2 is not trivial, but its Whitneysum with a 1-dimensional trivial bundle is trivial. Thus, S 2 is stably parallelizable.stationary set A set of ordinals S ⊆ κ whichmeets every closed unbounded set in κ; i.e., S ∩C = ∅ for each closed and unbounded C ⊆ κ.Stationary sets are somewhat large; for example,they are unbounded because for each α < κ,Cα = [α, κ) is closed and unbounded.Steenrod algebraThe algebra of all cohomology operations for ordinary mod p cohomology, for a prime p.
When p = 2, the Steenrod square operations Sq i generate the Steenrodalgebra. For odd primes p, the analog of thesquares are the pth power operations P i ; thesetogether with the Bockstein operation generatethe Steenrod algebra for p odd.The Steenrod squares are defined as additivecohomology operationsSq i : H q (X, A) −→ H q+i (X, A)(additive natural transformationsSq i : H q (−) −→ H q+i (−))© 2001 by CRC Press LLCThe power operations are defined by similar properties.
See also Bockstein operation.The cohomology of a space is a comoduleover the Steenrod algebra. This structure (coaction) is preserved by many (not all) constructions and calculational techniques, and hencecan be used to calculate the cohomology of certain spaces, for example, Eilenberg MacLanespaces (whose cohomology can be calculatedusing the Serre spectral sequence).Steenrod pth power operationalgebra.Steenrod square operationgebra.See SteenrodSee Steenrod al-stereographic projectionAn identificationof the plane with the sphere minus a point, N ,say, obtained by projecting from N a point Pon the sphere different from N .
If the sphereof radius 1 touches the (x, y) coordinate planeat the origin and N = (0, 0, 2), the projection2y2xsends (x, y, z) to ( 2−z, 2−z).Sometimes, instead, the entire sphere is identified with the complex plane, together with thepoint at infinity. See also complex sphere.Stirling number The number Snm for n ≥ m,which denotes the number of partitions of a setof n objects into m non-empty subsets. Thesenumbers are given by the recurrence relationskSn1 = 1 = Snn and Sn+1= Snk−1 + kSnk for1 < k < n.Stone-Čech compactificationThe uniquelargest compactification β(X) of a completelyregular topological space, X.
Its usefulness derives from the fact that any continuous functionfrom X to a compact Hausdorff space may beextended uniquely and continuously to β(X).strong inductionTo construct β(X), let F be the set of allcontinuous functions from X to the closed unitinterval, [0, 1]. Then the product space [0, 1]F ,of one copy of the unit interval for each f ∈ F,is a compact Hausdorff space by Tychonoff’sTheorem.
Imbed X in [0, 1]F by mapping x ∈X to the element of the product with f (x) in itsf -coordinate. β(X) is the closure of the imageof X under this imbedding. See also one-pointcompactification.strong inductionA method of proof overwell-ordered sets. In practice, strong inductionis typically used over the set of natural numbers.Strong induction has a base-case, like induction,but a different inductive step. Expressed in formal notation, the base-case is P (n0 ), for somen0 ; the inductive step has the form:(∀k)[[(∀n ≤ k)P (n)] → P (k + 1)] .From these the conclusion is (∀k ≥ n0 )P (k),where P (k) is some statement and n0 , n, k arenatural numbers.
Strong induction is equivalentto induction. See induction.strongly multiplicative functionA multiplicative function f having the property thatf (pi ) = f (p) for all primes p and all positiveintegers i. For example, the function f (n) =φ(n)n , where φ is the Euler phi function, is stronglymultiplicative. See multiplicative function, Euler phi function. See also completely multiplicative function.strong pseudoprimeSee pseudoprime.structureA mapping A, which assigns values to the quantifier symbol, the predicate symbols, the constant symbols, and the function symbols of a first order language L, as follows.(i.) A assigns to the quantifier symbol ∀ anonempty set A (sometimes denoted by |A|),called the universe of A.(ii.) For each n-ary predicate symbol P , Aassigns P to an n-ary relation P A ⊆ An .(iii.) For each constant symbol c, A assignsc to an element cA of A.(iv.) For each n-ary function symbol f , Aassigns f to an n-ary function f A : An → A.For example, if L is the language of elementary number theory (see first order language),© 2001 by CRC Press LLCthen one possible structure for L is the intendedstructure N , which assigns the quantifier ∀ to N,the set of natural numbers, and <, 0, S, +, ·, Eto their intended interpretations on N.
Thereare other (non-standard) structures for this language.A structure is sometimes called a model.subbasis for a topology A collection of subsets of a topological space X whose set of finiteintersections forms a basis for the topology τ ofX. For example, the set of all open intervals ofthe form (−∞, a) or (a, ∞) is a subbasis for theusual topology on R because each basic open set(a, b) can be written as (−∞, b) ∩ (a, ∞).Any collection S of subsets of a nonemptyset X generates a topology on X by declaring Sto be a subbasis. That is, the topology is the setof all unions of finite intersections of elementsfrom S.
The topology generated in this way isthe smallest topology on X which contains S.subbundleA bundle F −→ E −→ Bcontained a given bundle F −→ E −→ B.The tangent bundle and normal bundle of amanifold M embedded in Rn are both subbundles of the trivial bundle M × Rn .subcategoryC is a subcategory of a category C if (i.) every object of C is an object ofC, (ii.) for every pair of objects A, B of C , iff : A → B is a morphism of C , then f is amorphism of C, and (iii.) for every pair f, g ofmorphisms of C , the compositions f ◦c g andf ◦c g are the same morphisms in C and C . C isa full subcategory of C if, in addition, for everypair A, B of objects of C , f : A → B is a morphism of C if and only if f is a morphism ofC.
For example, the category of sets and bijective functions is a subcategory of the categoryof sets and injective functions; the category ofAbelian groups and group homomorphisms is afull subcategory of the category of groups andgroup homomorphisms.subobject If A is an object of a category C, asubobject of A is an ordered pair (f, A ), whereA is an object of C and f is a monomorphismf : A → A. For example, in the category ofgroups and group homomorphisms, a subobjectof the additive group Z is (f, E), where E issum of kth powers of divisors functionthe additive group of even integers and f is theinclusion map f : E → Z.
The dual notion ofsubobject is the notion of quotient object. Seequotient object.is α ∪ {α}; it is the least ordinal that is greaterthan α. For example, 3 + 1 = 4 and the ordinalsuccessor of ω is ω+1. Compare with successorof a cardinal.subset (of a set) A set S is a subset of a set X ifall elements of S are also elements of X. If S is asubset of X, the notation is S ⊆ X, or sometimesS ⊂ X. For example, {4, −2} ⊆ {−2, 5, 4}.Every set is a subset of itself. Compare withproper subset.successor of a setis S ∪ {S}.subspaceAny subset of a topological spaceX, with the relative topology inherited from X.See relative topology. For example, besides containing all its open subintervals, the subspacetopology on the unit interval [0, 1] also includesthe half-open intervals [0, b) with b ≤ 1 and(a, 1] with a ≥ 0.substructure The structure A for the first order language L is a substructure of the structureB for L (notation: A ⊆ B) if(i.) A ⊆ B, where A and B are the universesof A and B, respectively,(ii.) for each n-ary predicate symbol P , then-ary relation P A is the restriction of P B to An ;i.e., P A = P B ∩ An ,(iii.) for each constant symbol c, cA = cB ,and(iv.) for each n-ary function symbol f , f Ais the restriction of f B to An .If A is a substructure of B, then B is calledan extension of A.Sometimes the term submodel is synonymouswith substructure.successor cardinal A cardinal number κ suchthat there exists some other cardinal λ such thatλ+ = κ.
For example, ℵ17 is a successor cardinal since ℵ+16 = ℵ17 ; ℵ17 is a limit ordinal.Compare with limit cardinal, successor ordinal.successor of a cardinal If κ is a cardinal, thecardinal successor of κ, denoted by κ + , is theleast cardinal that is greater than κ. For example,3+ = 4 and ℵ+0 = ℵ1 . Compare with successorof an ordinal.successor of an ordinalIf α is an ordinal,the ordinal successor of α, denoted by α + 1,© 2001 by CRC Press LLCIf S is any set, its successorsuccessor ordinal An ordinal number α suchthat there exists some other ordinal β such thatβ + 1 = α.
For example, ω3 + 5 is a successorordinal since (ω3 + 4) + 1 = ω3 + 5. The cardinal number ℵ1 is a successor cardinal but nota successor ordinal. Compare with successorcardinal, limit ordinal.sum of cardinal numbers The cardinal number that is equinumerous with the disjoint unionof the summands. For example, 1 + 1 = 2, andℵ0 + ℵ3 = ℵ3 .sum of divisors function The arithmetic function, denoted σ , which, for any positive integern, returns thesum of the positive divisors of n,i.e., σ (n) = d|n d.
(See arithmetic function.)For example, σ (10) = 1 + 2 + 5 + 10 = 18.It is multiplicative; its value at a prime power isgiven byσ (p i ) =pi+1 − 1.p−1See also multiplicative function, sum of kth powers of divisors function.sum of kth powers of divisors functionThe family of arithmetic functions, denoted σk ,which, for any positive integer n and a fixednonnegative integer k, returns the sum of thekth powersof the positive divisors of n, i.e.,kσk (n) =d|n d . (See arithmetic function.)For example,σ2 (8) = 12 + 22 + 42 + 82 = 85 .The function σ0 is the number of divisors function τ , and σ1 is the sum of divisors functionσ .
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