Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 35
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Seesimply connected space. That is, D must be© 2001 by CRC Press LLCpath-connected and have a trivial fundamentalgroup, π1 (D), as a subspace of X.simply connected space A topological spaceX which is path-connected and has trivial fundamental group π1 (X). That is, any closed path inX is homotopic to a constant path. For example,a circular disc in the plane is simply connected,while an annulus is not because there are pathsin it (going around the annulus) which cannotbe continuously deformed to a constant path.singleton setAny set with exactly one element.
For example, {7} is a singleton.singular cardinal A cardinal number whosecofinality is smaller than itself. Thus, if κ is asingular cardinal, κ is not regular and cf(κ) <κ. For example, ℵω is a singular cardinal. Compare with regular cardinal.singular complex For X a topological space,the chain complex S(X) = {Sn (X)} of freeAbelian groups (or free modules over a ring R),generated by singular simplices. See chain complex. The standard k-simplex is the set σk ={(x0 , x1 , . . . , xk ) ∈ R k+1 : x0 + . . . + xk = 1,each xi ≥ 0}. A singular k-simplex is a continuous function φ : σk −→ X.
For eachn ≥ 0, Sn (X) is the free module generated bythe singular k-simplices. The boundary map∂k : Sk (X) −→ Sk−1 (X) is constructed by taking a singular simplex φ to the alternating sumof the (k − 1) − simplices determined by restricting φ to its faces.singular homology A graded Abelian groupH (X) = {Hn (X)}, determined by a space X.The group Hk (X) is the quotient of the singular cycles Zk (X) = ker∂k : Sk (X) −→Sk−1 (X) modulo the boundaries Bk (X) =∂k+1 (Sk+1 (X)). The singular homology groupsare fundamental invariants of X.singular n-boundaryIf {Sn (X)} is the singular complex of a space X, then the nth boundary group Bn (X) is the subgroup of Sn (X) consisting of elements of the form ∂c for c inSn+1 (X). The elements of Bn (X) are singularn-boundaries.smooth structuresingular n-chain An element of the free Abelian group (or, more generally, the free moduleover a ring R) Sn (X), a linear combination ofsingular n-simplices in a topological space X.See singular n-simplex.singular n-simplexis the setThe standard n-simplexσn = {(x0 , x1 , .
. . , xn ) ∈ Rn+1 :x0 + . . . + xn = 1, each xi ≥ 0} .A singular n-simplex in a space X is a continuous function φ : σn −→ X.skew linesTwo lines that do not meet inprojective geometry, which can occur in Pn forn ≥ 3 only.Skolem expansion(1) The Skolem expansion of a language L is L ∪ {fφ :∃xφ is a formula in L}, where each fφ is aSkolem function for φ. See Skolem function.(2) The Skolem expansion of a theory T inthe language L is T together with the set of sentences∀ȳ ∃xφ(x, ȳ) → φ(fφ (ȳ), ȳ)for each Skolem function fφ of L. The languageof the expanded theory is the Skolem expansionof L.(3) A Skolem expansion of a structure A inthe language L is a model A which adds to Aconsistent interpretations of the Skolem functions of L.
That is, for each Skolem function fφof L,A |= ∀ȳ ∃xφ(x, ȳ) → φ(fφ (ȳ), ȳ) .The language of the expanded model A is theSkolem expansion of L.Skolem functionIf ∃xφ(x, ȳ) is a formulawith all its free variables in ȳ = {y1 , . . . , yn },then a Skolem function for φ, fφ , satisfies∀ȳ ∃xφ(x, ȳ) → φ(fφ (ȳ), ȳ) .In effect, the symbol fφ (ȳ) names a witness ofthe existential statement ∃xφ(x, ȳ) for each ȳwhich has one.© 2001 by CRC Press LLCSkolem hull If X is a subset of an L-structureA, the Skolem hull of X is the smallest submodelof the Skolem expansion of A which contains X.Equivalently, it is the smallest subset of A containing X which is closed under the operations ofthe Skolem expansion.
Any nonempty Skolemhull is an elementary submodel of the originalmodel A. See Skolem expansion.Skolem normal formA formula ψ is inSkolem normal form if ψ = ∀x̄∃ȳφ(x̄, ȳ), whereφ is quantifier-free.Skolem theoryA theory T in the languageL which is its own Skolem expansion; that is, Tcontains∀ȳ ∃xφ(x, ȳ) → φ(fφ (ȳ), ȳ)for each Skolem function fφ of L.
See Skolemfunction.smoothing A smooth structure on a topological manifold M, which induces the given topological structure. A smoothing of a piecewiselinear manifold is a smooth structure in whicheach simplex is smooth.smoothing problemThe problem of determining the existence or non-existence of asmoothing of a topological or piecewise linearmanifold M.
See smoothing. The problem always has an affirmative solution in dimensionsless than or equal to three, but there are manycounterexamples in higher dimensions, both forexistence and uniqueness of smoothings.smooth manifoldA real manifold whosetransition functions are smooth, or C (k) differentiable, for k ≥ 1. Namely, a spaceM with an open covering {Uα } and identifications φα : Uα → Rn , where n is the dimension of the manifold and the transition functions φαβ : Uα ∩ Uβ → Uα ∩ Uβ are such thatφα = φβ ◦ φαβ where they are all defined.smooth structureA maximal collection oflocal coordinate systems on a manifold with theproperty that the coordinate transformation between any two overlapping coordinate systemsis differentiable with differentiable inverse.Sorgenfrey lineSorgenfrey line The real line with the topology given by taking the collection of all halfopen intervals [a, b) (or (a, b]) as a basis.
It isalso known as the lower (or upper) limit topology.The Sorgenfrey line is normal and Lindelöfbut not second countable. Its product with itself(the Sorgenfrey plane) is neither normal nor Lindelöf. Thus, it is an example showing that theproduct of normal spaces need not be normal,and the product of Lindelöf spaces need not beLindelöf. See normal space.Sorgenfrey planeSee Sorgenfrey line.space of complex numbersThe complexnumbers, visualized as a plane with real andimaginary axes, together with the usual (product) topology of the plane, is a topological space.The set of purely imaginary numbers forms asubspace homeomorphic to the real line.The imaginary axis, considered a subspace,is homeomorphic to the real numbers.space of imaginary numberscomplex numbers.See space ofspace of irrational numbersA subspaceof the space of real numbers: closeness, as described by open sets, is determined by open intervals in the real numbers intersected with therespective set.
The space is dense in the spaceof real numbers; that is, its closure is the spaceof real numbers.space of rational numbers A subspace, usually denoted Q or Q, of the space of real numbers: closeness, as described by open sets, is determined by open intervals in the real numbersintersected with the respective set. The space Qis dense in the space of real numbers; that is, itsclosure is the space of real numbers.space of real numbers The set of real numbers together with the usual real line topologygenerated by open intervals, usually denotedR, R, R1 or E1 .
Intuitively, open sets describecloseness, and typical uses of the real numbersrequire a topology where decreasing intervalsaround a point describe points strictly closer tothat point. R is also a metric space with dis-© 2001 by CRC Press LLCtance function d(x, y) = |x − y|. See also realnumber.span The smallest subspace of a vector spaceF containing a given set C of vectors in F .sphere (1) The subspace S n of Rn+1 consisting of all points (x1 , . . . , xn+1 ) with x12 + · · · +2= 1.xn+1(2) More generally, a space homeomorphicto S n .spherical distance The greatest lower boundof the lengths of all paths between two points pand q lying on the (unit) sphere.
It is the lengthof the short great circle arc joining p to q.spherical polygon A closed curve on the surface of the sphere made up of a finite number ofgreat circle arcs.spherical triangle A closed curve consistingof three points A, B, and C on the sphere, together with a great circle arc joining each pairof points. Sometimes the arcs are required to beshortest arcs.square-free integer An integer that is not divisible by any perfect square other than 1. Theprime factorization of a square-free integer contains no exponent greater than 1. Thus, 21 issquare-free, but 20 is not, since 22 is a divisorof 20.square numbersome integer n.An integer that equals n2 forsquare root (1) (Of a non-negative real number r) The unique non-negativereal number s so√that s 2 = r, denoted r.(2) If z and w are complex numbers such thatw2 = z, then w is said to be a square root of z(there will be two square roots of a given nonzerocomplex number, since if w is a square root ofz, so is −w and by the Fundamental Theoremof Algebra, the equation x 2 = z has at most twodistinct solutions).stable (primary) cohomology operation LetX denote the suspension of a space X (S 1 ∧X).Then H q (X) is isomorphic to H q+1 (X) byStone-Čech compactificationan isomorphism called the suspension isomorphism (here denoted ), natural in X.A cohomology operation P is stable whenP = P , that is, P commutes with the suspension isomorphism.
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