Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 3
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This is equivalentto the complete type of every ā being principal.Any finite model is atomic, as is the standardmodel of number theory.atom of a Boolean algebraIf (B, ∨, ∧,∼, 1, 0) is a Boolean algebra, a ∈ B is an atomif it is a minimal element of B\{0}. For exam-automorphismple, in the Boolean algebra of the power set ofany nonempty set, any singleton set is an atom.considered to be an axiom of logic, not an axiomof set theory.automorphismLet L be a first order language and let A be a structure for L.
An automorphism of A is an isomorphism from A ontoitself. See isomorphism.Axiom of Extensionality If two sets have thesame elements, then they are equal. This is oneof the axioms of Zermelo-Fraenkel set theory.axiomatic set theoryA collection of statements concerning set theory which can be provedfrom a collection of fundamental axioms. Thevalidity of the statements in the theory plays norole; rather, one is only concerned with the factthat they can be deduced from the axioms.Axiom of ChoiceSuppose that {Xα }α∈ isa family of non-empty, pairwise disjoint sets.Then there exists a set Y which consists of exactly one element from each set in the family.Equivalently, given any family of non-emptysets{Xα }α∈ , there exists a function f : {Xα }α∈→ α∈ Xα such that f (Xα ) ∈ Xα for eachα ∈ .The existence of such a set Y or function fcan be proved from the Zermelo-Fraenkel axioms when there are only finitely many sets inthe family.
However, when there are infinitelymany sets in the family it is impossible to provethat such Y, f exist or do not exist. Therefore,neither the Axiom of Choice nor its negation canbe proved from the axioms of Zermelo-Fraenkelset theory.Axiom of ComprehensionAlso called Axiom of Separation. See Axiom of Separation.Axiom of Constructibility Every set is constructible. See constructible set.Axiom of Dependent Choiceof dependent choices.Axiom of Infinity There exists an infinite set.This is one of the axioms of Zermelo-Fraenkelset theory.
See infinite set.Axiom of RegularityEvery non-empty sethas an ∈ -minimal element. More precisely, every non-empty set S contains an element x ∈ Swith the property that there is no element y ∈ Ssuch that y ∈ x. This is one of the axioms ofZermelo-Fraenkel set theory.Axiom of ReplacementIf f is a function,then, for every set X, there exists a set f (X) ={f (x) : x ∈ X}. This is one of the axioms ofZermelo-Fraenkel set theory.Axiom of Separation If P is a property andX is a set, then there exists a set Y = {x ∈ X : xsatisfies property P }.This is one of the axioms of Zermelo-Fraenkel set theory.
It is a weaker version of the Axiom of Comprehension: if P is a property, thenthere exists a set Y = {X : X satisfies propertyP }. Russell’s Paradox shows that the Axiom ofComprehension is false for sets. See also Russell’s Paradox.Axiom of SubsetsSame as the Axiom ofSeparation. See Axiom of Separation.See principleAxiom of DeterminancyFor any set X ⊆ωω , the game GX is determined. This axiomcontradicts the Axiom of Choice. See determined.Axiom of EqualityIf two sets are equal,then they have the same elements. This is theconverse of the Axiom of Extensionality and is© 2001 by CRC Press LLCAxiom of FoundationSame as the Axiomof Regularity. See Axiom of Regularity.Axiom of the Empty Set∅ which has no elements.There exists a setAxiom of the Power SetFor every set X,there exists a set P (X), the set of all subsets ofX.
This is one of the axioms of Zermelo-Fraenkel set theory.Axiom of the Unordered Pair If X and Y aresets, then there exists a set {X, Y }. This axiom,Axiom of Unionalso known as the Axiom of Pairing, is one ofthe axioms of Zermelo-Fraenkel set theory.© 2001 by CRC Press LLCAxiom of UnionFor any set S, there existsa set that is the union of all the elements of S.base of number systemBBaire classThe Baire classes Bα are an increasing sequence of families of functions defined inductively for α < ω1 . B0 is the set ofcontinuous functions. For α > 0, f is in Baireclass α if there is a sequence of functions {fn }converging pointwise to f , with fn ∈ Bβn andβn < α for each n.
Thus, f is in Baire class1 (or is Baire-1) if it is the pointwise limit ofa sequence of continuous functions. In somecases, it is useful to define the classes so that iff ∈ Bα , then f ∈/ Bβ for any β < α. See alsoBaire function.Baire functionA function belonging to oneof the Baire classes, Bα , for some α < ω1 .Equivalently, the set of Baire functions in a topological space is the smallest collection containing all continuous functions which is closed under pointwise limits. See Baire class.It is a theorem that f is a Baire function ifand only if f is Borel measurable, that is, if andonly if f −1 (U ) is a Borel set for any open setU.Baire measurable functionA function f :X → Y , where X and Y are topological spaces,such that the inverse image of any open set hasthe Baire property.
See Baire property. That is,if V ⊆ Y is open, thenf −1 (V ) = U C = (U \ C) ∪ (C \ U ) ,where U ⊆ X is open and C ⊆ X is meager.Baire propertyA set that can be written asan open set modulo a first category or meagerset. That is, X has the Baire property if there isan open set U and a meager set C withX = U C = (U \ C) ∪ (C \ U ) .Since the meager sets form a σ -ideal, this happens if and only if there is an open set U andmeager sets C and D with X = (U \ C) ∪ D.Every Borel set has the Baire property; in fact,every analytic set has the Baire property.© 2001 by CRC Press LLCBaire space(1) A topological space X suchthat no nonempty open set in X is meager (firstcategory).
That is, no open set U = ∅ in Xmay be written as a countable union of nowheredense sets. Equivalently, X is a Baire space ifand only if the intersection of any countable collection of dense open sets in X is dense, which istrue if and only if, for any countable collection ofclosed sets {Cn } with empty interior, their union∪Cn also has empty interior. The Baire Category Theorem states that any complete metricspace is a Baire space.(2) The Baire space is the set of all infinite sequences of natural numbers, NN , with the product topology and using the discrete topology oneach copy of N.
Thus, U is a basic open set inNN if there is a finite sequence of natural numbers σ such that U is the set of all infinite sequences which begin with σ . The Baire spaceis homeomorphic to the irrationals.bar constructionFor a group G, one canconstruct a space BG as the geometric realization of the following simplicial complex. Thefaces Fn in simplicial degree n are given by(n + 1)-tuples of elements of G.
The boundarymaps Fn −→ Fn−1 are given by the simplicialboundary formulan(−1)i (g0 , . . . , gˆi , . . . , gn )i=0where the notation gˆi indicates that gi is omittedto obtain an n-tuple. The ith degeneracy mapsi : Fn −→ Fn+1 is given by inserting the groupidentity element in the ith position.Example: B(Z/2), the classifying space ofthe group Z/2, is RP ∞ , real infinite projectivespace (the union of RP n for all n positive integers).The bar construction has many generalizations and is a useful means of constructing thenerve of a category or the classifying space of agroup, which determines the vector bundles ofa manifold with the group acting on the fiber.base of number systemThe number b, inuse, when a real number r is written in the formr=Nj =−∞rj b j ,Bernays-Gödel set theorywhere each rj = 0, 1, ..., b − 1, and r is represented in the notationr = rN rN−1 · · · r0 .r−1 r−2 · · · .For example, the base of the standard decimalsystem is 10 and we need the digits 0, 1, 2, 3,4, 5, 6, 7, 8, and 9 in order to use this system.Similarly, we use only the digits 0 and 1 in thebinary system; this is a “base 2” system.
Inthe base b system, the number 10215.2011 isequivalent to the decimal number1 × b4 + 0 × b3 + 2 × b2 + 1 × b + 5 + 2 × b−1+0 × b−2 + 1 × b−3 + 1 × b−4 .That is, each place represents a specific powerof the base b. See also radix.Bernays-Gödel set theory An axiomatic settheory, which is based on axioms other thanthose of Zermelo-Fraenkel set theory. BernaysGödel set theory considers two types of objects:sets and classes.
Every set is a class, but theconverse is not true; classes that are not setsare called proper classes. This theory has theAxioms of Infinity, Union, Power Set, Replacement, Regularity, and Unordered Pair for setsfrom Zermelo-Fraenkel set theory. It also hasthe following axioms, with classes written in :(i.) Axiom of Extensionality (for classes):Suppose that X and Y are two classes such thatU ∈ X if and only if U ∈ Y for all set U . ThenX = Y.(ii.) If X ∈ Y, then X is a set.(iii.) Axiom of Comprehension: For any formula F (X) having sets as variables there existsa class Y consisting of all sets satisfying the formula F (X).Bertrand’s postulateIf x is a real numbergreater than 1, then there is at least one primenumber p so that x < p < 2x.
Bertrand’s Postulate was conjectured to be true by the Frenchmathematician Joseph Louis Francois Bertrandand later proved by the Russian mathematicianPafnuty Lvovich Tchebychef.Betti numberSuppose X is a space whosehomology groups are finitely generated. Thenthe kth homology group is isomorphic to the direct sum of a torsion group Tk and a free Abelian© 2001 by CRC Press LLCgroup Bk .
The kth Betti number bk (X) of X isthe rank of Bk . Equivalently, bk (X) is the dimension of Hk (X, Q), the kth homology groupwith rational coefficients, viewed as a vectorspace over the rationals. For example, b0 (X)is the number of connected components of X.bijectionA function f : X → Y , betweentwo sets, with the following two properties:(i.) f is one-to-one (if x1 , x2 ∈ X and f (x1 )= f (x2 ), then x1 = x2 );(ii.) f is onto (for any y ∈ Y there exists anx ∈ X such that f (x) = y).See function.binomial coefficient(1) If n and k are nonnegative integerswithk≤n, then the binomial n!coefficient nk equals k!(n−k)!.
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