Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 34

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Then fora cardinal κ, A is κ-saturated if for any X ⊆A of size less than κ, every type (x) in thelanguage LX which is consistent with the theoryof A (using LX ) is realized in A. That is, thereis some a ∈ A such that A |= φ(a) for everyφ ∈ .A model A is saturated if it is |A|-saturated.The rationals are a saturated model of the theoryof dense linear orderings without endpoints.Schauder Fixed-Point Theorem Let X be aclosed convex subset of a Banach space. Thenany continuous map f : X → X for which theclosure of f (X) is compact must have a fixedpoint; that is, there is an x ∈ X with f (x) = x.In particular, any continuous mapping from acompact convex subset of a Banach space intoitself has a fixed point.Schröder-Bernstein TheoremBernstein Theorem.See Cantor-s-cobordismA geometric notion of equivalence for piecewise linear manifolds.

An hcobordism is a manifold W with boundary thedisjoint union of two manifolds M0 and M1 , inwhich the inclusion maps i0 : M0 −→ W andi1 : M1 −→ W are homotopy equivalences.This can be refined using the notion of simplehomotopy. If (K,L) is a pair of simplicial complexes with K = L ∪ B, with B a closed n-cells-Cobordism Theoremand B ∩ L is a face of B, then K is said to collapse to L, and L expands to K. This generatesan equivalence relation on polyhedra called simple homotopy equivalence. An s-cobordism isan h-cobordism in which the inclusions i0 andi1 are simple homotopy equivalences.s-Cobordism TheoremLet W be ans-cobordism, with boundary the disjoint unionof two manifolds M0 and M1 .

Then, if the dimension of W is at least 6, W is actually equivalent (as a polyhedral manifold) to the productmanifold M0 × [0, 1]. This would be false ifthe inclusion maps were only homotopy equivalences.secondary cohomology operationAn image of a lift of a cohomology class u in H i (Y ; A),formed in the following manner.

The class u isrepresented by a map u : Y −→ K(A, i). Letα be a cohomology operation corresponding tothe map α : K(A, i) −→ K(B, j ) for whichαu = 0. Let X represent the two-stage Postnikov tower given by α, so thatK(B, j − 1) −→ X −→ K(A, i) −→ K(B, j )is a fibration with maps i, p, and α, respectively.Let β : X −→ K(G, n) represent a class inH n (X; G). Since αu = 0, there is a map v suchthat v composed with the map X −→ K(A, i)is homotopic to u.

The cohomology class inH n (Y ; G) given by composing βu is the secondary cohomology operation given by this procedure evaluated on u.This operation is only determined up to acoset. If everything is done in the stable range,then the indeterminacy is due only to the choiceof v; any two choices may differ by any element of H n (Y, G) which is in the image ofi ∗ (α) : H j −1 (Y ; B) −→ H n (Y ; G). One usually only uses secondary operations in the stablerange (j and n less than 2i − 1) because indeterminacy is difficult to determine otherwise.These are operations that arise from the relations among primary cohomology operations.The Adem relation Sq 3 Sq 1 + Sq 2 Sq 2 = 0 generates a secondary cohomology operation thatshows that η2 is essential (not homotopic to zero),where η represents the Hopf map S 3 −→ S 2 (inthe Hopf bundle) or any suspension of that map.Note that a secondary cohomology operation is© 2001 by CRC Press LLCnot defined on the whole cohomology group ingeneral.

See Adem relations. See also primarycohomology operation.second category spaceA topological spaceX which is not first category; that is, X is notequal to the union of a countable collection ofclosed subsets with empty interiors.second countable topological space A topological space that has a countable basis for itstopology. For example, the real line (with itsstandard topology) is second countable sinceopen intervals with rational endpoints form abasis.semicircle An arc of a circle, connecting twopoints on a diameter, for example {(x, y) : x 2 +y 2 = 1, y ≥ 0}.sentenceA well-formed formula of a firstorder language having no free variables. Seefree variable.sentential calculuslus.sentential logicSee propositional calcu-See propositional logic.separable topological spaceA topologicalspace with a countable, dense subset.

For example, the real line (with its standard topology)is separable, since the set of rational numbers iscountable and dense in the reals.separated setsTwo subsets A and B of atopological space X which satisfy Ā ∩ B = B̄ ∩A = ∅, where Ā and B̄ denote the closure of Aand B.separation axiomsA system of axioms fortopological spaces X which measure, in increasing fashion, the extent to which points and subsets are separated by the topology on X. Thestandard axioms are called the T0 , T1 , T2 , T3 ,and T4 axioms. Other axioms, including completely regular, Tychonoff, and Urysohn spaces,refine and extend this list.separation by a continuous functionTheproperty of a continuous function f : X →sieve of Eratosthenes[0, 1] that, for two subsets A, B ⊂ X, we havef (A) = {0} and f (B) = {1}.set(1) In naive set theory.

A set is any collection of arbitrary objects. When such a collection is seen as a single entity, it is considereda set. Alternative terms: collection or family (inparticular, these terms are often used for sets ofsets, sets of sets of sets, and so on; so a set of setsis often called a family of sets, or a collectionof sets).

Sets are determined by their elements(their members). Standard set notation definesa set by listing or describing its elements withincurly brackets: {,}. For example, the set whoseonly elements are the number 3 and the letterQ is written in list form as {3, Q} or {Q, 3}(the order of listing does not matter). The set{2, 4, 6, 8} (list form) can also be expressed indescription form as {2n : n = 1, 2, 3, 4} or as{2n | n = 1, 2, 3, 4}.(2) In axiomatic set theory.

A formal mathematical object whose existence is a consequenceof the axiomatic system with which one is working. For example, in Zermelo-Fraenkel set theory, sets are built using axioms such as Union,Comprehension, Power Set, etc. See ZermeloFraenkel set theory, Bernays-Gödel set theory.set theory(1) Axiomatic set theory. Thebranch of mathematics whose purpose is to studysets within a formal axiomatic framework. Alsoknown as the foundation of mathematics, referring to the notion that all of mathematics canbe carried out within set theory.

For example,Zermelo-Fraenkel set theory models mathematics in a natural way. See also Zermelo-Fraenkelset theory, Bernays-Gödel set theory.(2) Naive set theory. The practice of dealing with sets as arbitrary collections of objectsand performing operations on such sets withoutappealing to axioms.sexagesimal number system A number system, used by the ancient Babylonian civilization,that was a base 60 positional system, in contrast to the base 10 positional system commonlyused today.

The value of a particular numberdepends both on the numerals used in its representation and the placement of these numerals.Using the symbols | and < to represent 1 and10 respectively, one can denote the number 34© 2001 by CRC Press LLCas <<< ||||, 154 is expressed as || <<< ||||(2 × 60 + 34), and 5000 is represented by | <<||| << ((1×602 )+(23×60)+20). A high basesuch as 60 is useful for dealing with large numbers, since the “place values” represent powersof the base (60), namely, 1, 60, 3600, 216000,. .

. One of the difficulties is the fact that theremust be 60 “digits” (representing the values 0to 59). In fact, the Babylonians did not have asymbol for zero so the number << ||||||| < |||could represent 1663 ((27 × 60) + 13) or 97213(27 × 3600) + (0 × 60) + 13). See also base ofnumber system.sheafA structure F on a topological spaceX, which assigns an object F (U ) to each opensubset U of X, and for each inclusion U in Vof open sets in X, F assigns a restriction maprV ,U : F (V ) −→ F (U ) so that rU,U is theidentity on F (U ) and whenever U in V in W arenested open sets, rV ,U ◦ rW,V = rW,U . Further,whenever U = ∪a∈I Ua is a covering of U byopen sets Ua , and {fa } is a collection of elementsfa in F (Ua ) such that the restrictions of fa andfb to Ua ∩Ub are equal, there is a unique elementf in F (U ) such that the restriction rU,Ua (f ) toeach Ua is just fa .Example: The collection of open sets of aspace X is a sheaf with F (U ) = U .

One mayalso use sheaves as coefficients in homology ofX.sieve of Eratosthenes A method (named after the Greek mathematician Eratosthenes) for“sifting” out the prime numbers less than a fixedinteger N . It relies on the fact that if n is a positive integer less than or equal to N , then n iseither a prime number or√ has a prime factor thatis less than or equal to N .To find the primes less than or equal to N ,first list the integers from 2 to N . Then, circle2 and cross out all of the other multiples of 2since we know they cannot be primes (they aredivisible by 2). Notice that the smallest integerleft that is not circled or crossed out is 3 (thenext prime number). Circle 3 and cross out theremaining multiples of 3.

Now, circle the smallest integer that is neither circled nor crossed out(5) and cross out all its other multiples. Continue this process until the smallest integer thatis neither circled nor crossed out is greater thansimple closed curve√N . Circle the remaining integers in the list;the integers that have been circled are the primesless than or equal to N .simple closed curveA topological space Cthat is homeomorphic to the unit circle. Intuitively, this means that C does not cross itself.simple homotopy equivalence A homotopyequivalence f : S1 → S2 between two simplicial complexes which is obtained as a composition of elementary contractions and expansions.Given an n-simplex σ of a simplicial complex Ssuch that σ is the face of a unique n + 1-simplexτ , an elementary contraction of S is the map thatcollapses σ and τ to a point.

An elementaryexpansion of S is the inverse of an elementarycontraction.simplexLet {a0 , . . . , an } be a geometrically independent subset of Rn . The n-simplexσ spanned by {a0 , . . . , an } is the set of all pointsx=ntk ak , wherek=0ntk = 1 ,k=0and tk ≥ 0 for all k. The points {a0 , . . . , an }are called the vertices of σ. The tk are calledthe barycentric coordinates for σ. Any simplexspanned by a subset of {a0 , . . . , an } is called aface of σ. For example, a 0-simplex is a point, a1-simplex is a line segment, and a 2-simplex isa triangle.simplicial approximationLet f : S1 →S2 be a continuous function between simplicialcomplexes. A simplicial mapping g : S1 → S2is a simplicial approximation for f if f (St(v)) ⊆St(g(v)) for every vertex v of S1 where St(v) denotes the star of the vertex v.simplicial complexA set V of vertices, together with a set K of finite subsets of V calledsimplices, satisfying the condition: if σ is a simplex and τ is a subset of σ , then τ is also a simplex.simply connected domainA subset D of atopological space X which is open, connected,and simply connected as a subspace of X.

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