Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 37
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The functions σk are all multiplicative; theirvalue at a prime power is given byσk (p i ) =pk(i+1) − 1.pk − 1surdsurdAnother name for the radical signSee also radical.√.surjectionA function f : A → B such thatthe image (range) of f is all of B; that is, forany b ∈ B there is an a ∈ A with f (a) = b.Suslin lineA dense linear ordering (L, <)which in the order topology has the countablechain condition but is not separable. That is, Lhas no countable dense subset, but any collectionof pairwise disjoint nonempty open sets in L iscountable.It is possible to characterize the real line Ras the unique dense linear order without endpoints which is complete and separable.
Thequestion arose as to whether separability couldbe replaced by the countable chain condition,and so the existence of a Suslin line would meanthat this new set of conditions does not characterize R. However, the existence of a Suslin lineis independent of the axioms of set theory, andthus, so is the characterization. See also Suslintree.Suslin’s hypothesisThe assertion that thereis no Suslin line. See Suslin line. That is, thereis no dense linear ordering which in the ordertopology has the countable chain condition butis not separable. Souslin’s hypothesis (abbreviated SH) is independent of the axioms of set© 2001 by CRC Press LLCtheory: it is a consequence of Martin’s Axiom,but ¬SH is a consequence of Diamond (♦), whichholds in the constructible universe.
These results are usually obtained indirectly, by considering Suslin trees rather than Suslin lines. Seealso Martin’s Axiom.Suslin treeA tree of height ω1 , which hasno uncountable antichains or branches. That is,any subset A ⊆ T consisting of incomparableelements (antichain) or any set B ⊆ T totallyordered by < (branch) must be countable. Theexistence of Suslin trees is independent of theaxioms of set theory. In fact, Suslin trees provide a way to prove the independence of Suslin’shypothesis (SH) because a Suslin tree exists ifand only if a Suslin line exists. See Suslin line.For any infinite cardinal κ, a κ-Suslin tree isa tree of height κ in which all antichains andbranches have size less than κ.See also Aronszajn tree, Kurepa tree.symmetric difference The symmetric difference of two sets A and B, written AB, is theset (A \ B) ∪ (B \ A).
That is, it is the set ofall elements that belong to either A or B but notboth.symmetric relation A binary relation R suchthat (x, y) ∈ R implies (y, x) ∈ R, for all x, y.For example, the equality relation is symmetric.terminal objectTT0 spaceA topological space X such that,for any two distinct points of X, there is a neighborhood of one which does not contain the other.That is, for all x, y in X, with x = y, there is anopen set U such that either x ∈ U and y ∈/ U,or y ∈ U and x ∈/ U . T0 spaces are also knownas Kolmogorov spaces.A topological space X such that,T1 spacefor any two distinct points of X, there are neighborhoods of both which do not contain the other.That is, for all x, y in X, with x = y, there areopen sets U and V such that x ∈ U and y ∈/ U,while y ∈ V and x ∈/ V .
This is equivalent toeach singleton {x} being closed in X.T2 space A topological space X such that anytwo distinct points can be separated by open sets.That is, for all x, y in X, with x = y, there areopen sets U and V such that x ∈ U , y ∈ V ,and U ∩ V = ∅. See also Hausdorff topologicalspace.T3 1 spaceA topological space X such that2X is a T1 space and points and closed sets in Xcan be separated by continuous functions. Thatis, for all closed C ⊆ X and x ∈/ C, there is acontinuous f : X → [0, 1] such that f (x) = 0and f (c) = 1 for all c ∈ C.
Including the condition T1 ensures T3 1 ⊆ T3 . See also completely2regular topological space, T1 space.A topological space X which is aT3 spaceT1 space and such that points and closed sets canbe separated by open sets. That is, for all closedC ⊆ X and x ∈/ C, there are open sets U andV such that x ∈ U , C ⊆ V , and U ∩ V = ∅.Including the condition T1 ensures T3 ⊆ T2 . Seealso regular topological space, T1 space.A topological space X which is aT4 spaceT1 space and such that disjoint closed sets can beseparated by open sets.
That is, for all closed Cand D contained in X, if C ∩ D = ∅, then there© 2001 by CRC Press LLCare open sets U and V such that C ⊆ U , D ⊆ V ,and U ∩ V = ∅. Including the condition T1ensures T4 ⊆ T3 1 . See also normal topological2space, T1 space.A topological space X which is aT5 spaceT1 space and such that any two separated setscan be separated by disjoint open sets. That is,for all subsets A and B of X, ifA∩B =A∩B =∅(A and B are separated), then there are open setsU and V with A ⊆ U , B ⊆ V , and U ∩ V = ∅.Including the condition T1 ensures T5 ⊆ T4 . Forany T1 space X, T5 is equivalent to hereditarynormality. See also T1 space.tautology In propositional (sentential) logic,a well-formed propositional formula is a tautology if it is true under every truth assignment forthe sentence symbols in the formula. For example, if A and B are sentence symbols, then¬(A ∨ B) ↔ ((¬A) ∧ (¬B)),(which is one of DeMorgan’s Laws) is a tautology.In first order logic, let L be a first order language.
A tautology is any well-formed formulaof L which is obtained from a tautology of propositional logic by replacing each sentence symbolin the tautology with a well-formed formula ofL.term Let L be a first order language. The setof terms of L is defined inductively as follows.(i.) If c is a constant symbol of L, then c is aterm.(ii.) If v is a variable of L, then v is a term.(iii.) If f is an n-place function symbol of Land t1 , . . . , tn are terms of L, then f (t1 , .
. . , tn )is a term of L.(iv.) The set of terms is generated by rules(i.), (ii.), and (iii.).For example, in the first order language ofelementary number theory (see first order language), S(0) is a term (which is intended to namethe natural number 1).terminal object An object A of a category Csuch that, for every object B of C, there is exactlyterminating decimalone morphism f of C such that f : B → A. Forexample, in the category of sets and functions, asingleton is a terminal object.
The dual notionof terminal object is initial object.terminating decimaltionA decimal representa-. . . a4 a3 a2 a1 a0 .a−1 a−2 a−3 . . .of a real number such that there is an integer Nwith a−n = 0 for all n ≥ N . A real number rhas a terminating decimal representation if andonly if there is an integer a and a nonnegativeinteger N so that r = 10aN . Clearly, any realnumber with a terminating decimal representation is therefore a rational number.ternary number systemThe real numbersin base b = 3 notation.
See base of numbersystem.theorem In first order logic, let L be a first order language, and consider a particular predicatecalculus for L. Let α be a well-formed formulaof L. Then α is a theorem of (or is deduciblefrom) the predicate calculus (notation: α) ifthere is a proof of α in the predicate calculus.See proof. If is a set of well-formed formulas of L, then α is a theorem of (or is deduciblefrom) (in the predicate calculus) if there is aproof of α from (notation: α).The notion of theorem in propositional logicis entirely analogous.theoryA set T of sentences of a first orderlanguage L which is closed under logical implication; i.e., if σ is a sentence of L which is alogical consequence of T , then σ ∈ T (in notation, T |= σ implies σ ∈ T ).
Equivalently, Tis a theory if it is closed under deduction; i.e., ifσ is provable from T , then σ ∈ T (in notation,T σ implies σ ∈ T ).For some authors, the word theory simplymeans a set of sentences, and the notion aboveis that of a closed theory.Let A be a structure for L. The theory of Ais the set of sentences of L which are true in A(i.e., the theory of A is the set of sentences σsuch that A is a model of σ ). The theory of Ais denoted T h(A) and is a complete theory. Seecomplete theory.© 2001 by CRC Press LLCThom complex Let E −→ M be a real vector bundle on a manifold M.
There is a disk bundle D −→ M which is given by the open unitdisk in each fiber of the vector bundle E. TheThom construction is formed from E −→ Mby identifying all points in E outside of D to asingle point, called the point at infinity.Example: Consider the Möbius band as a linebundle over the circle S 1 . The Thom complex ofthis bundle is the real projective plane.This construction is used in the calculationof cobordism groups. See R. Stong, Notes onCobordism Theory, Princeton University Press,Princeton, NJ, 1968.topological dimensionLet X be a topological space. The topological dimension of X isthe smallest non-negative integer n such that, forevery open cover A of X, there is an open coverB that refines A (i.e., A ⊆ B), with the propertythat some point of X lies in an element of B andno point of X lies in more than n + 1 elementsof B.topological group A topological space whichis also a group such that the inverse and product maps are both continuous.
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