# Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics

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DICTIONARY OF

Classical

AND

Theoretical

mathematics

© 2001 by CRC Press LLC

a Volume in the

Comprehensive Dictionary

of Mathematics

DICTIONARY OF

Classical

AND

Theoretical

mathematics

Edited by

Catherine Cavagnaro

William T. Haight, II

CRC Press

Boca Raton London New York Washington, D.C.

© 2001 by CRC Press LLC

Preface

The Dictionary of Classical and Theoretical Mathematics, one volume of the Comprehensive

Dictionary of Mathematics, includes entries from the fields of geometry, logic, number theory,

set theory, and topology. The authors who contributed their work to this volume are professional

mathematicians, active in both teaching and research.

The goal in writing this dictionary has been to define each term rigorously, not to author a

large and comprehensive survey text in mathematics. Though it has remained our purpose to make

each definition self-contained, some definitions unavoidably depend on others, and a modicum of

“definition chasing” is necessitated. We hope this is minimal.

The authors have attempted to extend the scope of this dictionary to the fringes of commonly

accepted higher mathematics. Surely, some readers will regard an excluded term as being mistakenly overlooked, and an included term as one “not quite yet cooked” by years of use by a broad

mathematical community. Such differences in taste cannot be circumnavigated, even by our wellintentioned and diligent authors. Mathematics is a living and breathing entity, changing daily, so a

list of included terms may be regarded only as a snapshot in time.

We thank the authors who spent countless hours composing original definitions. In particular, the

help of Dr. Steve Benson, Dr. William Harris, and Dr. Tamara Hummel was key in organizing the

collection of terms. Our hope is that this dictionary becomes a valuable source for students, teachers,

researchers, and professionals.

Catherine Cavagnaro

William T. Haight, II

© 2001 by CRC Press LLC

© 2001 by CRC Press LLC

CONTRIBUTORS

Curtis Bennett

Krystyna Kuperberg

Bowling Green State University

Bowling Green, Ohio

Auburn University

Steve Benson

Thomas LaFramboise

University of New Hampshire

Durham, New Hampshire

Marietta College

Catherine Cavagnaro

University of the South

Sewanee, Tennessee

Auburn, Alabama

Marietta, Ohio

Adam Lewenberg

University of Akron

Akron, Ohio

Minevra Cordero

Texas Tech University

Lubbock, Texas

Elena Marchisotto

California State University

Northridge, California

Douglas E. Ensley

Shippensburg University

Shippensburg, Pennsylvania

William T. Haight, II

Rick Miranda

Colorado State University

Fort Collins, Colorado

University of the South

Sewanee, Tennessee

Emma Previato

William Harris

Boston, Massachusetts

Georgetown College

Georgetown, Kentucky

V.V. Raman

Boston University

Rochester Institute of Technology

Phil Hotchkiss

Pittsford, New York

University of St. Thomas

St. Paul, Minnesota

David A. Singer

Case Western Reserve University

Matthew G. Hudelson

Cleveland, Ohio

Washington State University

Pullman, Washington

David Smead

Tamara Hummel

Allegheny College

Meadville, Pennsylvania

Furman University

Greenville, South Carolina

Sam Smith

Mark J. Johnson

St. Joseph’s University

Central College

Pella, Iowa

Philadelphia, Pennsylvania

Paul Kapitza

Allegheny College

Illinois Wesleyan University

Bloomington, Illinois

Meadville, Pennsylvania

© 2001 by CRC Press LLC

Vonn Walter

Jerome Wolbert

Olga Yiparaki

University of Michigan

Ann Arbor, Michigan

University of Arizona

Tucson, Arizona

© 2001 by CRC Press LLC

absolute value

abscissa of convergence

For the Dirichlet

?

f (n)

series

ns , the real number ?c , if it exists,

A

n=1

Abelian category

An additive category C,

which satisfies the following conditions, for any

morphism f ? HomC (X, Y ):

(i.) f has a kernel (a morphism i ? HomC

(X , X) such that f i = 0) and a co-kernel (a

morphism p ? HomC (Y, Y ) such that pf = 0);

(ii.) f may be factored as the composition of

an epic (onto morphism) followed by a monic

(one-to-one morphism) and this factorization is

unique up to equivalent choices for these morphisms;

(iii.) if f is a monic, then it is a kernel; if f

is an epic, then it is a co-kernel.

See additive category.

Abel’s summation identity

If a(n) is an

arithmetical function (a real or complex valued

function defined on the natural numbers), define

A(x) =

0

a(n)

n?x

if x < 1 ,

if x ? 1 .

If the function f is continuously differentiable

on the interval [w, x], then

a(n)f (n)

=

A(x)f (x)

w ?a

but not for any s so that x < ?a . If the series

converges absolutely for all s, then ?a = ??

and if the series fails to converge absolutely for

any s, then ?a = ?. The set {x + iy : x > ?a }

is called the half plane of absolute convergence

for the series. See also abscissa of convergence.

© 2001 by CRC Press LLC

such that the series converges for all complex

numbers s = x + iy with x > ?c but not for

any s so that x < ?c . If the series converges

absolutely for all s, then ?c = ?? and if the

series fails to converge absolutely for any s, then

?c = ?. The abscissa of convergence of the

series is always less than or equal to the abscissa

of absolute convergence (?c ? ?a ). The set

{x + iy : x > ?c } is called the half plane of

convergence for the series. See also abscissa of

absolute convergence.

absolute neighborhood retract

A topological space W such that, whenever (X, A) is a

pair consisting of a (Hausdorff) normal space

X and a closed subspace A, then any continuous function f : A ?? W can be extended

to a continuous function F : U ?? W , for

U some open subset of X containing A. Any

absolute retract is an absolute neighborhood retract (ANR). Another example of an ANR is the

n-dimensional sphere, which is not an absolute

retract.

absolute retract A topological space W such

that, whenever (X, A) is a pair consisting of a

(Hausdorff) normal space X and a closed subspace A, then any continuous function f : A ??

W can be extended to a continuous function

F : X ?? W . For example, the unit interval

is an absolute retract; this is the content of the

Tietze Extension Theorem. See also absolute

neighborhood retract.

absolute value

quantity

(1) If r is a real number, the

r

if r ? 0 ,

?r

if r < 0 .

?

Equivalently, |r| = r 2 . For example, | ? 7|

= |7| = 7 and | ? 1.237| = 1.237. Also called

magnitude of r.

(2) If z = x + iy is a complex number, then

|z|, also referred

to as the norm or modulus of

2 + y 2 . For example, |1 ? 2i| =

z,

equals

x

?

?

12 + 22 = 5.

(3) In Rn (Euclidean n space), the absolute

value of an element is its (Euclidean) distance

|r| =

abundant number

to the origin. That is,

|(a1 , a2 , . . . , an )| =

a12 + a22 + · · · + an2 .

In particular, if a is a real or complex number,

then |a| is the distance from a to 0.

abundant number A positive integer n having the property that the sum of its positive divisors is greater than 2n, i.e., ? (n) > 2n. For

example, 24 is abundant, since

1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 > 48 .

additive functor

An additive functor F :

C ? D, between two additive categories, such

that F (f + g) = F (f ) + F (g) for any f, g ?

HomC (A, B). See additive category, functor.

Adem relations The relations in the Steenrod

algebra which describe a product of pth power

or square operations as a linear combination of

products of these operations. For the square operations (p = 2), when 0 < i < 2j ,

Sq i Sq j =

0?k?[i/2]

j ?k?1

i ? 2k

Sq i+j ?k Sq k ,

The smallest odd abundant number is 945. Compare with deficient number, perfect number.

accumulation point A point x in a topological space X such that every neighborhood of x

contains a point of X other than x. That is, for all

open U ? X with x ? U , there is a y ? U which

is different from x. Equivalently, x ? X \ {x}.

More generally, x is an accumulation point

of a subset A ? X if every neighborhood of x

contains a point of A other than x. That is, for

all open U ? X with x ? U , there is a y ?

U ? A which is different from x. Equivalently,

x ? A \ {x}.

additive category A category C with the following properties:

(i.) the Cartesian product of any two elements of Obj(C) is again in Obj(C);

(ii.) HomC (A, B) is an additive Abelian group

with identity element 0, for any A, B ?Obj(C);

(iii.) the distributive laws f (g1 + g2 ) =

f g1 + f g1 and (f1 + f2 )g = f1 g + f2 g hold for

morphisms when the compositions are defined.

See category.

additive function

An arithmetic function f

having the property that f (mn) = f (m) + f (n)

whenever m and n are relatively prime. (See

arithmetic function). For example, ?, the number of distinct prime divisors function, is additive. The values of an additive function depend only on its values at powers of primes: if

n = p1i1 · · · pkik and f is additive, then f (n) =

f (p1i1 ) + . . . + f (pkik ). See also completely additive function.

© 2001 by CRC Press LLC

where [i/2] is the greatest integer less than or

equal to i/2 and the binomial coefficients in the

sum are taken mod 2, since the square operations

are a Z/2-algebra.

As a consequence of the values of the binomial coefficients, Sq 2n?1 Sq n = 0 for all values

of n.

The relations for Steenrod algebra of pth

power operations are similar.

adjoint functor

If X is a fixed object in a

category X , the covariant functor Hom? : X ?

Sets maps A ?Obj (X ) to HomX (X, A); f ?

HomX (A, A ) is mapped to f? : HomX (X, A)

? HomX (X, A ) by g ? f g. The contravariant functor Hom? : X ? Sets maps A ?Obj(X )

to HomX (A, X); f ? HomX (A, A ) is mapped

to

f ? : HomX (A , X) ? HomX (A, X) ,

by g ? gf .

Let C, D be categories. Two covariant functors F : C ? D and G : D ? C are adjoint

functors if, for any A, A ? Obj(C), B, B ?

Obj(D), there exists a bijection

? : HomC (A, G(B)) ? HomD (F (A), B)

that makes the following diagrams commute for

any f : A ? A in C, g : B ? B in D:

algebraic variety

HomC (A,

? G(B))

?

?

HomD (F (A), B)

HomC (A,

? G(B))

?

?

HomD (F (A), B)

f?

??

(F (f ))?

HomC (A? , G(B))

?

?

??

HomD (F (A ), B)

(G(g))?

??

HomC (A,

? G(B ))

?

?

g?

HomD (F (A), B )

??

See category of sets.

alephs Form the sequence of infinite cardinal

numbers (?? ), where ? is an ordinal number.

Alexander’s Horned Sphere An example of

a two sphere in R3 whose complement in R3 is

not topologically equivalent to the complement

of the standard two sphere S 2 ? R3 .

This space may be constructed as follows:

On the standard two sphere S 2 , choose two mutually disjoint disks and extend each to form two

“horns” whose tips form a pair of parallel disks.

On each of the parallel disks, form a pair of

horns with parallel disk tips in which each pair

of horns interlocks the other and where the distance between each pair of horn tips is half the

previous distance. Continuing this process, at

stage n, 2n pairwise linked horns are created.

In the limit, as the number of stages of the

construction approaches infinity, the tips of the

horns form a set of limit points in R3 homeomorphic to the Cantor set. The resulting surface is

homeomorphic to the standard two sphere S 2 but

the complement in R3 is not simply connected.

algebra of sets A collection of subsets S of a

non-empty set X which contains X and is closed

with respect to the formation of finite unions,

intersections, and differences. More precisely,

(i.) X ? S;

(ii.) if A, B ? S, then A ? B, A ? B, and

A\B are also in S.

See union, difference of sets.

algebraic number

(1) A complex number

which is a zero of a polynomial with rational coefficients (i.e., ? is algebraic if there exist ratio-

© 2001 by CRC Press LLC

Alexander’s Horned Sphere.

PovRay.

Graphic rendered by

n

nal numbers a0 , a1 , . . . , an so that

ai ? i = 0).

i=0

?

For example, 2 is an algebraic number since

it satisfies the equation x 2 ? 2 = 0. Since there

is no polynomial p(x) with rational coefficients

such that p(? ) = 0, we see that ? is not an algebraic number. A complex number that is not

an algebraic number is called a transcendental

number.

(2) If F is a field, then ? is said to be algebraic over F if ? is a zero of a polynomial

having coefficients in F . That is, if there exist

elements f0 , f1 , f2 , . . . , fn of F so that f0 +

f1 ? + f2 ? 2 · · · + fn ? n = 0, then ? is algebraic

over F .

algebraic number field

A subfield of the

complex numbers consisting entirely of algebraic numbers. See also algebraic number.

algebraic number theory

That branch of

mathematics involving the study of algebraic

numbers and their generalizations. It can be argued that the genesis of algebraic number theory

was Fermat’s Last Theorem since much of the

results and techniques of the subject sprung directly or indirectly from attempts to prove the

Fermat conjecture.

algebraic variety Let A be a polynomial ring

k[x1 , . . . , xn ] over a field k. An affine algebraic

variety is a closed subset of An (in the Zariski

topology of An ) which is not the union of two

proper (Zariski) closed subsets of An . In the

Zariski topology, a closed set is the set of common zeros of a set of polynomials. Thus, an

affine algebraic variety is a subset of An which

is the set of common zeros of a set of polynomi-

altitude

als but which cannot be expressed as the union

of two such sets.

The topology on an affine variety is inherited

from An .

In general, an (abstract) algebraic variety is a

topological space with open sets Ui whose union

is the whole space and each of which has an

affine algebraic variety structure so that the induced variety structures (from Ui and Uj ) on

each intersection Ui ? Uj are isomorphic.

The solutions to any polynomial equation form

an algebraic variety. Real and complex projective spaces can be described as algebraic varieties (k is the field of real or complex numbers,

respectively).

altitude

In plane geometry, a line segment

joining a vertex of a triangle to the line through

the opposite side and perpendicular to the line.

The term is also used to describe the length of

the line segment. The area of a triangle is given

by one half the product of the length of any side

and the length of the corresponding altitude.

amicable pair of integers

Two positive integers m and n such that the sum of the positive

divisors of both m and n is equal to the sum of

m and n, i.e., ? (m) = ? (n) = m + n. For

example, 220 and 284 form an amicable pair,

since

? (220) = ? (284) = 504 .

A perfect number forms an amicable pair with

itself.

analytic number theory That branch of mathematics in which the methods and ideas of real

and complex analysis are applied to problems

concerning integers.

analytic set The continuous image of a Borel

set. More precisely, if X is a Polish space and

A ? X, then A is analytic if there is a Borel set B

contained in a Polish space Y and a continuous

f : X ? Y with f (A) = B. Equivalently, A

is analytic if it is the projection in X of a closed

set

C ? X ? NN ,

where NN is the Baire space. Every Borel set is

analytic, but there are analytic sets that are not

© 2001 by CRC Press LLC

Borel. The collection of analytic sets is denoted

11 . See also Borel set, projective set.

annulus A topological space homeomorphic

to the product of the sphere S n and the closed

unit interval I . The term sometimes refers specifically to a closed subset of the plane bounded by

two concentric circles.

antichain

A subset A of a partially ordered

set (P , ?) such that any two distinct elements

x, y ? A are not comparable under the ordering

?. Symbolically, neither x ? y nor y ? x for

any x, y ? A.

arc

A subset of a topological space, homeomorphic to the closed unit interval [0, 1].

arcwise connected component If p is a point

in a topological space X, then the arcwise connected component of p in X is the set of points

q in X such that there is an arc (in X) joining

p to q. That is, for any point q distinct from

p in the arc component of p there is a homeomorphism ? : [0, 1] ?? J of the unit interval

onto some subspace J containing p and q. The

arcwise connected component of p is the largest

arcwise connected subspace of X containing p.

arcwise connected topological space A topological space X such that, given any two distinct

points p and q in X, there is a subspace J of X

homeomorphic to the unit interval [0, 1] containing both p and q.

arithmetical hierarchy A method of classifying the complexity of a set of natural numbers

based on the quantifier complexity of its definition. The arithmetical hierarchy consists of

classes of sets n0 , 0n , and 0n , for n ? 0.

A set A is in 00 = 00 if it is recursive (computable). For n ? 1, a set A is in n0 if there is

a computable (recursive) (n + 1)–ary relation R

such that for all natural numbers x,

x ? A ?? (?y1 )(?y2 ) . . . (Qn yn )R(x, y),

where Qn is ? if n is odd and Qn is ? if n is

odd, and where y abbreviates y1 , . . . , yn . For

n ? 1, a set A is in 0n if there is a computable

(recursive) (n + 1)–ary relation R such that for

atom of a Boolean algebra

all natural numbers x,

x ? A ?? (?y1 )(?y2 ) . . . (Qn yn )R(x, y),

where Qn is ? if n is even and Qn is ? if n is

odd. For n ? 0, a set A is in 0n if it is in both

n0 and 0n .

Note that it suffices to define the classes n0

and 0n as above since, using a computable coding function, pairs of like quantifiers (for example, (?y1 )(?y2 )) can be contracted to a single

quantifier ((?y)). The superscript 0 in n0 , 0n ,

0n is sometimes omitted and indicates classes

in the arithmetical hierarchy, as opposed to the

analytical hierarchy.

A set A is arithmetical if it belongs to the

arithmetical hierarchy; i.e., if, for some n, A

is in n0 or 0n . For example, any computably

(recursively) enumerable set is in 10 .

arithmetical set

A set A which belongs to

the arithmetical hierarchy; i.e., for some n, A

is in n0 or 0n . See arithmetical hierarchy. For

example, any computably (recursively) enumerable set is in 10 .

arithmetic function

A function whose domain is the set of positive integers. Usually, an

arithmetic function measures some property of

an integer, e.g., the Euler phi function ? or the

sum of divisors function ? . The properties of

the function itself, such as its order of growth or

whether or not it is multiplicative, are often studied. Arithmetic functions are also called number

theoretic functions.

Aronszajn tree

A tree of height ?1 which

has no uncountable branches or levels. Thus,

for each ? < ?1 , the ?-level of T , Lev? (T ),

given by

t ? T : ordertype({s ? T : s < t}) = ?

is countable, Lev?1 (T ) is the first empty level of

T , and any set B ? T which is totally ordered

by < (branch) is countable. An Aronszajn tree

is constructible in ZFC without any extra settheoretic hypotheses.

For any regular cardinal ?, a ?-Aronszajn tree

is a tree of height ? in which all levels have size

less than ? and all branches have length less than

?. See also Suslin tree, Kurepa tree.

© 2001 by CRC Press LLC

associated fiber bundle

A concept in the

theory of fiber bundles. A fiber bundle ? consists of a space B called the base space, a space

E called the total space, a space F called the

fiber, a topological group G of transformations

of F , and a map ? : E ?? B. There is a

covering of B by open sets Ui and homeomorphisms ?i : Ui ? F ?? Ei = ? ?1 (Ui ) such

that ? ? ?i (x, V ) = x. This identifies ? ?1 (x)

with the fiber F . When two sets Ui and Uj overlap, the two identifications are related by coordinate transformations gij (x) of F , which are

required to be continuously varying elements of

G. If G also acts as a group of transformations

on a space F , then the associated fiber bundle

? = ? : E ?? B is the (uniquely determined) fiber bundle with the same base space

B, fiber F , and the same coordinate transformations as ? .

associated principal fiber bundle The associated fiber bundle, of a fiber bundle ? , with the

fiber F replaced by the group G. See associated

fiber bundle. The group acts by left multiplication, and the coordinate transformations gij are

the same as those of the bundle ? .

atomic formula

Let L be a first order language. An atomic formula is an expression

which has the form P (t1 , . . . , tn ), where P is

an n-place predicate symbol of L and t1 , . . . , tn

are terms of L. If L contains equality (=), then

= is viewed as a two-place predicate. Consequently, if t1 and t2 are terms, then t1 = t2 is an

atomic formula.

atomic model

A model A in a language L

such that every n-tuple of elements of A satisfies a complete formula in T , the theory of

A. That is, for any a? ? An , there is an Lformula ? (x)

? such that A |= ? (a),

? and for any

L-formula

?,

either

T

?

x

?

?

(

x)

?

? ?(x)

? or

T ?x? ? (x)

? ? ¬?(x)

? . This is equivalent

to the complete type of every a? being principal.

Any finite model is atomic, as is the standard

model of number theory.

atom of a Boolean algebra

If (B, ?, ?,

?, 1, 0) is a Boolean algebra, a ? B is an atom

if it is a minimal element of B\{0}. For exam-

automorphism

ple, in the Boolean algebra of the power set of

any nonempty set, any singleton set is an atom.

considered to be an axiom of logic, not an axiom

of set theory.

automorphism

Let L be a first order language and let A be a structure for L. An automorphism of A is an isomorphism from A onto

itself. See isomorphism.

Axiom of Extensionality If two sets have the

same elements, then they are equal. This is one

of the axioms of Zermelo-Fraenkel set theory.

axiomatic set theory

A collection of statements concerning set theory which can be proved

from a collection of fundamental axioms. The

validity of the statements in the theory plays no

role; rather, one is only concerned with the fact

that they can be deduced from the axioms.

Axiom of Choice

Suppose that {X? }?? is

a 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-empty

sets

{X? }?? , there exists a function f : {X? }??

? ??

X? such that f (X? ) ? X? for each

? ? .

The existence of such a set Y or function f

can be proved from the Zermelo-Fraenkel axioms when there are only finitely many sets in

the family. However, when there are infinitely

many sets in the family it is impossible to prove

that such Y, f exist or do not exist. Therefore,

neither the Axiom of Choice nor its negation can

be proved from the axioms of Zermelo-Fraenkel

set theory.

Axiom of Comprehension

Also called Axiom of Separation. See Axiom of Separation.

Axiom of Constructibility Every set is constructible. See constructible set.

Axiom of Dependent Choice

of dependent choices.

Axiom of Infinity There exists an infinite set.

This is one of the axioms of Zermelo-Fraenkel

set theory. See infinite set.

Axiom of Regularity

Every non-empty set

has an ? -minimal element. More precisely, every non-empty set S contains an element x ? S

with the property that there is no element y ? S

such that y ? x. This is one of the axioms of

Zermelo-Fraenkel set theory.

Axiom of Replacement

If 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 of

Zermelo-Fraenkel set theory.

Axiom of Separation If P is a property and

X is a set, then there exists a set Y = {x ? X : x

satisfies 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, then

there exists a set Y = {X : X satisfies property

P }. Russell’s Paradox shows that the Axiom of

Comprehension is false for sets. See also Russell’s Paradox.

Axiom of Subsets

Same as the Axiom of

Separation. See Axiom of Separation.

See principle

Axiom of Determinancy

For any set X ?

?? , the game GX is determined. This axiom

contradicts the Axiom of Choice. See determined.

Axiom of Equality

If two sets are equal,

then they have the same elements. This is the

converse of the Axiom of Extensionality and is

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Axiom of Foundation

Same as the Axiom

of Regularity. See Axiom of Regularity.

Axiom of the Empty Set

? which has no elements.

There exists a set

Axiom of the Power Set

For every set X,

there exists a set P (X), the set of all subsets of

X. This is one of the axioms of Zermelo-Fraenkel set theory.

Axiom of the Unordered Pair If X and Y are

sets, then there exists a set {X, Y }. This axiom,

Axiom of Union

also known as the Axiom of Pairing, is one of

the axioms of Zermelo-Fraenkel set theory.

© 2001 by CRC Press LLC

Axiom of Union

For any set S, there exists

a set that is the union of all the elements of S.

base of number system

B

Baire class

The Baire classes B? are an increasing sequence of families of functions defined inductively for ? < ?1 . B0 is the set of

continuous functions. For ? > 0, f is in Baire

class ? if there is a sequence of functions {fn }

converging pointwise to f , with fn ? B?n and

?n < ? for each n. Thus, f is in Baire class

1 (or is Baire-1) if it is the pointwise limit of

a sequence of continuous functions. In some

cases, it is useful to define the classes so that if

f ? B? , then f ?

/ B? for any ? < ?. See also

Baire function.

Baire function

A function belonging to one

of the Baire classes, B? , for some ? < ?1 .

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 if

and only if f is Borel measurable, that is, if and

only if f ?1 (U ) is a Borel set for any open set

U.

Baire measurable function

A function f :

X ? Y , where X and Y are topological spaces,

such that the inverse image of any open set has

the Baire property. See Baire property. That is,

if V ? Y is open, then

f ?1 (V ) = U C = (U \ C) ? (C \ U ) ,

where U ? X is open and C ? X is meager.

Baire property

A set that can be written as

an open set modulo a first category or meager

set. That is, X has the Baire property if there is

an open set U and a meager set C with

X = U C = (U \ C) ? (C \ U ) .

Since the meager sets form a ? -ideal, this happens if and only if there is an open set U and

meager 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 LLC

Baire space

(1) A topological space X such

that no nonempty open set in X is meager (first

category). That is, no open set U = ? in X

may be written as a countable union of nowhere

dense sets. Equivalently, X is a Baire space if

and only if the intersection of any countable collection of dense open sets in X is dense, which is

true if and only if, for any countable collection of

closed sets {Cn } with empty interior, their union

?Cn also has empty interior. The Baire Category Theorem states that any complete metric

space 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 on

each copy of N. Thus, U is a basic open set in

NN if there is a finite sequence of natural numbers ? such that U is the set of all infinite sequences which begin with ? . The Baire space

is homeomorphic to the irrationals.

bar construction

For a group G, one can

construct a space BG as the geometric realization of the following simplicial complex. The

faces Fn in simplicial degree n are given by

(n + 1)-tuples of elements of G. The boundary

maps Fn ?? Fn?1 are given by the simplicial

boundary formula

n

(?1)i (g0 , . . . , g?i , . . . , gn )

i=0

where the notation g?i indicates that gi is omitted

to obtain an n-tuple. The ith degeneracy map

si : Fn ?? Fn+1 is given by inserting the group

identity element in the ith position.

Example: B(Z/2), the classifying space of

the group Z/2, is RP ? , real infinite projective

space (the union of RP n for all n positive integers).

The bar construction has many generalizations and is a useful means of constructing the

nerve of a category or the classifying space of a

group, which determines the vector bundles of

a manifold with the group acting on the fiber.

base of number system

The number b, in

use, when a real number r is written in the form

r=

N

j =??

rj b j ,

Bernays-G?del set theory

where each rj = 0, 1, ..., b ? 1, and r is represented in the notation

r = rN rN?1 · · · r0 .r?1 r?2 · · · .

For example, the base of the standard decimal

system 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 the

binary system; this is a “base 2” system. In

the base b system, the number 10215.2011 is

equivalent to the decimal number

1 ? 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 power

of the base b. See also radix.

Bernays-G?del set theory An axiomatic set

theory, which is based on axioms other than

those of Zermelo-Fraenkel set theory. BernaysG?del set theory considers two types of objects:

sets and classes. Every set is a class, but the

converse is not true; classes that are not sets

are called proper classes. This theory has the

Axioms of Infinity, Union, Power Set, Replacement, Regularity, and Unordered Pair for sets

from Zermelo-Fraenkel set theory. It also has

the following axioms, with classes written in :

(i.) Axiom of Extensionality (for classes):

Suppose that X and Y are two classes such that

U ? X if and only if U ? Y for all set U . Then

X = Y.

(ii.) If X ? Y, then X is a set.

(iii.) Axiom of Comprehension: For any formula F (X) having sets as variables there exists

a class Y consisting of all sets satisfying the formula F (X).

Bertrand’s postulate

If x is a real number

greater than 1, then there is at least one prime

number p so that x < p < 2x. Bertrand’s Postulate was conjectured to be true by the French

mathematician Joseph Louis Francois Bertrand

and later proved by the Russian mathematician

Pafnuty Lvovich Tchebychef.

Betti number

Suppose X is a space whose

homology groups are finitely generated. Then

the kth homology group is isomorphic to the direct sum of a torsion group Tk and a free Abelian

© 2001 by CRC Press LLC

group Bk . The kth Betti number bk (X) of X is

the rank of Bk . Equivalently, bk (X) is the dimension of Hk (X, Q), the kth homology group

with rational coefficients, viewed as a vector

space over the rationals. For example, b0 (X)

is the number of connected components of X.

bijection

A function f : X ? Y , between

two 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 an

x ? X such that f (x) = y).

See function.

binomial coefficient

(1) If n and k are nonnegative integers

with

k

?

n, then the binomial

n!

coefficient nk equals k!(n?k)!

.

(2) The binomial coefficient nk also represents the number of ways to choose k distinct

items from among n distinct items, without regard to the order of choosing.

(3)The binomial coefficient nk is the kth entry in the nth row of Pascal’s Triangle. It must be

noted that Pascal’s Triangle begins with row 0,

and each row begins with entry 0. See Pascal’s

triangle.

Binomial Theorem

If a and b are elements

of a commutative ring and

in

n is a non-negative

teger, then (a + b)n = nk=0 nk a k bn?k , where

n

k is the binomial coefficient. See binomial coefficient.

Bockstein operation In cohomology theory,

a cohomology operation is a natural transformation between two cohomology functors. If

0 ? A ? B ? C ? 0 is a short exact sequence of modules over a ring R, and if X ? Y

are topological spaces, then there is a long exact

sequence in cohomology:

· · · ? H q (X, Y ; A) ? H q (X, Y ; B) ?

H q (X, Y ; C) ?

H q+1 (X, Y ; A) ? H q+1 (X, Y ; B) ? . . . .

The homomorphism

? : H q (X, Y ; C) ? H q+1 (X, Y ; A)

is the Bockstein (cohomology) operation.

bounded quantifier

Bolzano-Weierstrass Theorem

Every

bounded sequence in R has a convergent subsequence. That is, if

{xn : n ? N} ? [a, b]

is an infinite sequence, then there is an increasing sequence {nk : k ? N} ? N such that

{xnk : k ? N} converges.

Boolean algebra

A non-empty set X, along

with two binary operations ? and ? (called union

and intersection, respectively), a unary operation (called complement), and two elements

0, 1 ? X which satisfy the following properties

for all A, B, C ? X.

(i.) A ? (B ? C) = (A ? B) ? C

(ii.) A ? (B ? C) = (A ? B) ? C

(iii.) A ? B = B ? A

(iv.) A ? B = B ? A

(v.) A ? (B ? C) = (A ? B) ? (A ? C)

(vi.) A ? (B ? C) = (A ? B) ? (A ? C)

(vii.) A ? 0 = A and A ? 1 = A

(viii.) There exists an element A so that A ?

A = 1 and A ? A = 0.

Borel measurable function

A function f :

X ? Y , for X, Y topological spaces, such that

the inverse image of any open set is a Borel set.

This is equivalent to requiring the inverse image

of any Borel set to be Borel. Any continuous

function is Borel measurable.

It is a theorem that f is Borel measurable

if and only if f is a Baire function. See Baire

function.

Borel set

The collection B of Borel sets of

a topological space X is the smallest ? -algebra

containing all open sets of X. That is, in addition

to containing open sets, B must be closed under

complements and countable intersections (and,

thus, is also closed under countable unions). For

comparison, the topology on X is closed under

arbitrary unions but only finite intersections.

Borel sets may also be defined inductively:

let 10 denote the collection of open sets and 01

the closed sets. Then for 1 < ? < ?1 , A ? ?0

if and only if

of B is in ?0 . Then the collection of all Borel

sets is

B = ??

Classical

AND

Theoretical

mathematics

© 2001 by CRC Press LLC

a Volume in the

Comprehensive Dictionary

of Mathematics

DICTIONARY OF

Classical

AND

Theoretical

mathematics

Edited by

Catherine Cavagnaro

William T. Haight, II

CRC Press

Boca Raton London New York Washington, D.C.

© 2001 by CRC Press LLC

Preface

The Dictionary of Classical and Theoretical Mathematics, one volume of the Comprehensive

Dictionary of Mathematics, includes entries from the fields of geometry, logic, number theory,

set theory, and topology. The authors who contributed their work to this volume are professional

mathematicians, active in both teaching and research.

The goal in writing this dictionary has been to define each term rigorously, not to author a

large and comprehensive survey text in mathematics. Though it has remained our purpose to make

each definition self-contained, some definitions unavoidably depend on others, and a modicum of

“definition chasing” is necessitated. We hope this is minimal.

The authors have attempted to extend the scope of this dictionary to the fringes of commonly

accepted higher mathematics. Surely, some readers will regard an excluded term as being mistakenly overlooked, and an included term as one “not quite yet cooked” by years of use by a broad

mathematical community. Such differences in taste cannot be circumnavigated, even by our wellintentioned and diligent authors. Mathematics is a living and breathing entity, changing daily, so a

list of included terms may be regarded only as a snapshot in time.

We thank the authors who spent countless hours composing original definitions. In particular, the

help of Dr. Steve Benson, Dr. William Harris, and Dr. Tamara Hummel was key in organizing the

collection of terms. Our hope is that this dictionary becomes a valuable source for students, teachers,

researchers, and professionals.

Catherine Cavagnaro

William T. Haight, II

© 2001 by CRC Press LLC

© 2001 by CRC Press LLC

CONTRIBUTORS

Curtis Bennett

Krystyna Kuperberg

Bowling Green State University

Bowling Green, Ohio

Auburn University

Steve Benson

Thomas LaFramboise

University of New Hampshire

Durham, New Hampshire

Marietta College

Catherine Cavagnaro

University of the South

Sewanee, Tennessee

Auburn, Alabama

Marietta, Ohio

Adam Lewenberg

University of Akron

Akron, Ohio

Minevra Cordero

Texas Tech University

Lubbock, Texas

Elena Marchisotto

California State University

Northridge, California

Douglas E. Ensley

Shippensburg University

Shippensburg, Pennsylvania

William T. Haight, II

Rick Miranda

Colorado State University

Fort Collins, Colorado

University of the South

Sewanee, Tennessee

Emma Previato

William Harris

Boston, Massachusetts

Georgetown College

Georgetown, Kentucky

V.V. Raman

Boston University

Rochester Institute of Technology

Phil Hotchkiss

Pittsford, New York

University of St. Thomas

St. Paul, Minnesota

David A. Singer

Case Western Reserve University

Matthew G. Hudelson

Cleveland, Ohio

Washington State University

Pullman, Washington

David Smead

Tamara Hummel

Allegheny College

Meadville, Pennsylvania

Furman University

Greenville, South Carolina

Sam Smith

Mark J. Johnson

St. Joseph’s University

Central College

Pella, Iowa

Philadelphia, Pennsylvania

Paul Kapitza

Allegheny College

Illinois Wesleyan University

Bloomington, Illinois

Meadville, Pennsylvania

© 2001 by CRC Press LLC

Vonn Walter

Jerome Wolbert

Olga Yiparaki

University of Michigan

Ann Arbor, Michigan

University of Arizona

Tucson, Arizona

© 2001 by CRC Press LLC

absolute value

abscissa of convergence

For the Dirichlet

?

f (n)

series

ns , the real number ?c , if it exists,

A

n=1

Abelian category

An additive category C,

which satisfies the following conditions, for any

morphism f ? HomC (X, Y ):

(i.) f has a kernel (a morphism i ? HomC

(X , X) such that f i = 0) and a co-kernel (a

morphism p ? HomC (Y, Y ) such that pf = 0);

(ii.) f may be factored as the composition of

an epic (onto morphism) followed by a monic

(one-to-one morphism) and this factorization is

unique up to equivalent choices for these morphisms;

(iii.) if f is a monic, then it is a kernel; if f

is an epic, then it is a co-kernel.

See additive category.

Abel’s summation identity

If a(n) is an

arithmetical function (a real or complex valued

function defined on the natural numbers), define

A(x) =

0

a(n)

n?x

if x < 1 ,

if x ? 1 .

If the function f is continuously differentiable

on the interval [w, x], then

a(n)f (n)

=

A(x)f (x)

w ?a

but not for any s so that x < ?a . If the series

converges absolutely for all s, then ?a = ??

and if the series fails to converge absolutely for

any s, then ?a = ?. The set {x + iy : x > ?a }

is called the half plane of absolute convergence

for the series. See also abscissa of convergence.

© 2001 by CRC Press LLC

such that the series converges for all complex

numbers s = x + iy with x > ?c but not for

any s so that x < ?c . If the series converges

absolutely for all s, then ?c = ?? and if the

series fails to converge absolutely for any s, then

?c = ?. The abscissa of convergence of the

series is always less than or equal to the abscissa

of absolute convergence (?c ? ?a ). The set

{x + iy : x > ?c } is called the half plane of

convergence for the series. See also abscissa of

absolute convergence.

absolute neighborhood retract

A topological space W such that, whenever (X, A) is a

pair consisting of a (Hausdorff) normal space

X and a closed subspace A, then any continuous function f : A ?? W can be extended

to a continuous function F : U ?? W , for

U some open subset of X containing A. Any

absolute retract is an absolute neighborhood retract (ANR). Another example of an ANR is the

n-dimensional sphere, which is not an absolute

retract.

absolute retract A topological space W such

that, whenever (X, A) is a pair consisting of a

(Hausdorff) normal space X and a closed subspace A, then any continuous function f : A ??

W can be extended to a continuous function

F : X ?? W . For example, the unit interval

is an absolute retract; this is the content of the

Tietze Extension Theorem. See also absolute

neighborhood retract.

absolute value

quantity

(1) If r is a real number, the

r

if r ? 0 ,

?r

if r < 0 .

?

Equivalently, |r| = r 2 . For example, | ? 7|

= |7| = 7 and | ? 1.237| = 1.237. Also called

magnitude of r.

(2) If z = x + iy is a complex number, then

|z|, also referred

to as the norm or modulus of

2 + y 2 . For example, |1 ? 2i| =

z,

equals

x

?

?

12 + 22 = 5.

(3) In Rn (Euclidean n space), the absolute

value of an element is its (Euclidean) distance

|r| =

abundant number

to the origin. That is,

|(a1 , a2 , . . . , an )| =

a12 + a22 + · · · + an2 .

In particular, if a is a real or complex number,

then |a| is the distance from a to 0.

abundant number A positive integer n having the property that the sum of its positive divisors is greater than 2n, i.e., ? (n) > 2n. For

example, 24 is abundant, since

1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 > 48 .

additive functor

An additive functor F :

C ? D, between two additive categories, such

that F (f + g) = F (f ) + F (g) for any f, g ?

HomC (A, B). See additive category, functor.

Adem relations The relations in the Steenrod

algebra which describe a product of pth power

or square operations as a linear combination of

products of these operations. For the square operations (p = 2), when 0 < i < 2j ,

Sq i Sq j =

0?k?[i/2]

j ?k?1

i ? 2k

Sq i+j ?k Sq k ,

The smallest odd abundant number is 945. Compare with deficient number, perfect number.

accumulation point A point x in a topological space X such that every neighborhood of x

contains a point of X other than x. That is, for all

open U ? X with x ? U , there is a y ? U which

is different from x. Equivalently, x ? X \ {x}.

More generally, x is an accumulation point

of a subset A ? X if every neighborhood of x

contains a point of A other than x. That is, for

all open U ? X with x ? U , there is a y ?

U ? A which is different from x. Equivalently,

x ? A \ {x}.

additive category A category C with the following properties:

(i.) the Cartesian product of any two elements of Obj(C) is again in Obj(C);

(ii.) HomC (A, B) is an additive Abelian group

with identity element 0, for any A, B ?Obj(C);

(iii.) the distributive laws f (g1 + g2 ) =

f g1 + f g1 and (f1 + f2 )g = f1 g + f2 g hold for

morphisms when the compositions are defined.

See category.

additive function

An arithmetic function f

having the property that f (mn) = f (m) + f (n)

whenever m and n are relatively prime. (See

arithmetic function). For example, ?, the number of distinct prime divisors function, is additive. The values of an additive function depend only on its values at powers of primes: if

n = p1i1 · · · pkik and f is additive, then f (n) =

f (p1i1 ) + . . . + f (pkik ). See also completely additive function.

© 2001 by CRC Press LLC

where [i/2] is the greatest integer less than or

equal to i/2 and the binomial coefficients in the

sum are taken mod 2, since the square operations

are a Z/2-algebra.

As a consequence of the values of the binomial coefficients, Sq 2n?1 Sq n = 0 for all values

of n.

The relations for Steenrod algebra of pth

power operations are similar.

adjoint functor

If X is a fixed object in a

category X , the covariant functor Hom? : X ?

Sets maps A ?Obj (X ) to HomX (X, A); f ?

HomX (A, A ) is mapped to f? : HomX (X, A)

? HomX (X, A ) by g ? f g. The contravariant functor Hom? : X ? Sets maps A ?Obj(X )

to HomX (A, X); f ? HomX (A, A ) is mapped

to

f ? : HomX (A , X) ? HomX (A, X) ,

by g ? gf .

Let C, D be categories. Two covariant functors F : C ? D and G : D ? C are adjoint

functors if, for any A, A ? Obj(C), B, B ?

Obj(D), there exists a bijection

? : HomC (A, G(B)) ? HomD (F (A), B)

that makes the following diagrams commute for

any f : A ? A in C, g : B ? B in D:

algebraic variety

HomC (A,

? G(B))

?

?

HomD (F (A), B)

HomC (A,

? G(B))

?

?

HomD (F (A), B)

f?

??

(F (f ))?

HomC (A? , G(B))

?

?

??

HomD (F (A ), B)

(G(g))?

??

HomC (A,

? G(B ))

?

?

g?

HomD (F (A), B )

??

See category of sets.

alephs Form the sequence of infinite cardinal

numbers (?? ), where ? is an ordinal number.

Alexander’s Horned Sphere An example of

a two sphere in R3 whose complement in R3 is

not topologically equivalent to the complement

of the standard two sphere S 2 ? R3 .

This space may be constructed as follows:

On the standard two sphere S 2 , choose two mutually disjoint disks and extend each to form two

“horns” whose tips form a pair of parallel disks.

On each of the parallel disks, form a pair of

horns with parallel disk tips in which each pair

of horns interlocks the other and where the distance between each pair of horn tips is half the

previous distance. Continuing this process, at

stage n, 2n pairwise linked horns are created.

In the limit, as the number of stages of the

construction approaches infinity, the tips of the

horns form a set of limit points in R3 homeomorphic to the Cantor set. The resulting surface is

homeomorphic to the standard two sphere S 2 but

the complement in R3 is not simply connected.

algebra of sets A collection of subsets S of a

non-empty set X which contains X and is closed

with respect to the formation of finite unions,

intersections, and differences. More precisely,

(i.) X ? S;

(ii.) if A, B ? S, then A ? B, A ? B, and

A\B are also in S.

See union, difference of sets.

algebraic number

(1) A complex number

which is a zero of a polynomial with rational coefficients (i.e., ? is algebraic if there exist ratio-

© 2001 by CRC Press LLC

Alexander’s Horned Sphere.

PovRay.

Graphic rendered by

n

nal numbers a0 , a1 , . . . , an so that

ai ? i = 0).

i=0

?

For example, 2 is an algebraic number since

it satisfies the equation x 2 ? 2 = 0. Since there

is no polynomial p(x) with rational coefficients

such that p(? ) = 0, we see that ? is not an algebraic number. A complex number that is not

an algebraic number is called a transcendental

number.

(2) If F is a field, then ? is said to be algebraic over F if ? is a zero of a polynomial

having coefficients in F . That is, if there exist

elements f0 , f1 , f2 , . . . , fn of F so that f0 +

f1 ? + f2 ? 2 · · · + fn ? n = 0, then ? is algebraic

over F .

algebraic number field

A subfield of the

complex numbers consisting entirely of algebraic numbers. See also algebraic number.

algebraic number theory

That branch of

mathematics involving the study of algebraic

numbers and their generalizations. It can be argued that the genesis of algebraic number theory

was Fermat’s Last Theorem since much of the

results and techniques of the subject sprung directly or indirectly from attempts to prove the

Fermat conjecture.

algebraic variety Let A be a polynomial ring

k[x1 , . . . , xn ] over a field k. An affine algebraic

variety is a closed subset of An (in the Zariski

topology of An ) which is not the union of two

proper (Zariski) closed subsets of An . In the

Zariski topology, a closed set is the set of common zeros of a set of polynomials. Thus, an

affine algebraic variety is a subset of An which

is the set of common zeros of a set of polynomi-

altitude

als but which cannot be expressed as the union

of two such sets.

The topology on an affine variety is inherited

from An .

In general, an (abstract) algebraic variety is a

topological space with open sets Ui whose union

is the whole space and each of which has an

affine algebraic variety structure so that the induced variety structures (from Ui and Uj ) on

each intersection Ui ? Uj are isomorphic.

The solutions to any polynomial equation form

an algebraic variety. Real and complex projective spaces can be described as algebraic varieties (k is the field of real or complex numbers,

respectively).

altitude

In plane geometry, a line segment

joining a vertex of a triangle to the line through

the opposite side and perpendicular to the line.

The term is also used to describe the length of

the line segment. The area of a triangle is given

by one half the product of the length of any side

and the length of the corresponding altitude.

amicable pair of integers

Two positive integers m and n such that the sum of the positive

divisors of both m and n is equal to the sum of

m and n, i.e., ? (m) = ? (n) = m + n. For

example, 220 and 284 form an amicable pair,

since

? (220) = ? (284) = 504 .

A perfect number forms an amicable pair with

itself.

analytic number theory That branch of mathematics in which the methods and ideas of real

and complex analysis are applied to problems

concerning integers.

analytic set The continuous image of a Borel

set. More precisely, if X is a Polish space and

A ? X, then A is analytic if there is a Borel set B

contained in a Polish space Y and a continuous

f : X ? Y with f (A) = B. Equivalently, A

is analytic if it is the projection in X of a closed

set

C ? X ? NN ,

where NN is the Baire space. Every Borel set is

analytic, but there are analytic sets that are not

© 2001 by CRC Press LLC

Borel. The collection of analytic sets is denoted

11 . See also Borel set, projective set.

annulus A topological space homeomorphic

to the product of the sphere S n and the closed

unit interval I . The term sometimes refers specifically to a closed subset of the plane bounded by

two concentric circles.

antichain

A subset A of a partially ordered

set (P , ?) such that any two distinct elements

x, y ? A are not comparable under the ordering

?. Symbolically, neither x ? y nor y ? x for

any x, y ? A.

arc

A subset of a topological space, homeomorphic to the closed unit interval [0, 1].

arcwise connected component If p is a point

in a topological space X, then the arcwise connected component of p in X is the set of points

q in X such that there is an arc (in X) joining

p to q. That is, for any point q distinct from

p in the arc component of p there is a homeomorphism ? : [0, 1] ?? J of the unit interval

onto some subspace J containing p and q. The

arcwise connected component of p is the largest

arcwise connected subspace of X containing p.

arcwise connected topological space A topological space X such that, given any two distinct

points p and q in X, there is a subspace J of X

homeomorphic to the unit interval [0, 1] containing both p and q.

arithmetical hierarchy A method of classifying the complexity of a set of natural numbers

based on the quantifier complexity of its definition. The arithmetical hierarchy consists of

classes of sets n0 , 0n , and 0n , for n ? 0.

A set A is in 00 = 00 if it is recursive (computable). For n ? 1, a set A is in n0 if there is

a computable (recursive) (n + 1)–ary relation R

such that for all natural numbers x,

x ? A ?? (?y1 )(?y2 ) . . . (Qn yn )R(x, y),

where Qn is ? if n is odd and Qn is ? if n is

odd, and where y abbreviates y1 , . . . , yn . For

n ? 1, a set A is in 0n if there is a computable

(recursive) (n + 1)–ary relation R such that for

atom of a Boolean algebra

all natural numbers x,

x ? A ?? (?y1 )(?y2 ) . . . (Qn yn )R(x, y),

where Qn is ? if n is even and Qn is ? if n is

odd. For n ? 0, a set A is in 0n if it is in both

n0 and 0n .

Note that it suffices to define the classes n0

and 0n as above since, using a computable coding function, pairs of like quantifiers (for example, (?y1 )(?y2 )) can be contracted to a single

quantifier ((?y)). The superscript 0 in n0 , 0n ,

0n is sometimes omitted and indicates classes

in the arithmetical hierarchy, as opposed to the

analytical hierarchy.

A set A is arithmetical if it belongs to the

arithmetical hierarchy; i.e., if, for some n, A

is in n0 or 0n . For example, any computably

(recursively) enumerable set is in 10 .

arithmetical set

A set A which belongs to

the arithmetical hierarchy; i.e., for some n, A

is in n0 or 0n . See arithmetical hierarchy. For

example, any computably (recursively) enumerable set is in 10 .

arithmetic function

A function whose domain is the set of positive integers. Usually, an

arithmetic function measures some property of

an integer, e.g., the Euler phi function ? or the

sum of divisors function ? . The properties of

the function itself, such as its order of growth or

whether or not it is multiplicative, are often studied. Arithmetic functions are also called number

theoretic functions.

Aronszajn tree

A tree of height ?1 which

has no uncountable branches or levels. Thus,

for each ? < ?1 , the ?-level of T , Lev? (T ),

given by

t ? T : ordertype({s ? T : s < t}) = ?

is countable, Lev?1 (T ) is the first empty level of

T , and any set B ? T which is totally ordered

by < (branch) is countable. An Aronszajn tree

is constructible in ZFC without any extra settheoretic hypotheses.

For any regular cardinal ?, a ?-Aronszajn tree

is a tree of height ? in which all levels have size

less than ? and all branches have length less than

?. See also Suslin tree, Kurepa tree.

© 2001 by CRC Press LLC

associated fiber bundle

A concept in the

theory of fiber bundles. A fiber bundle ? consists of a space B called the base space, a space

E called the total space, a space F called the

fiber, a topological group G of transformations

of F , and a map ? : E ?? B. There is a

covering of B by open sets Ui and homeomorphisms ?i : Ui ? F ?? Ei = ? ?1 (Ui ) such

that ? ? ?i (x, V ) = x. This identifies ? ?1 (x)

with the fiber F . When two sets Ui and Uj overlap, the two identifications are related by coordinate transformations gij (x) of F , which are

required to be continuously varying elements of

G. If G also acts as a group of transformations

on a space F , then the associated fiber bundle

? = ? : E ?? B is the (uniquely determined) fiber bundle with the same base space

B, fiber F , and the same coordinate transformations as ? .

associated principal fiber bundle The associated fiber bundle, of a fiber bundle ? , with the

fiber F replaced by the group G. See associated

fiber bundle. The group acts by left multiplication, and the coordinate transformations gij are

the same as those of the bundle ? .

atomic formula

Let L be a first order language. An atomic formula is an expression

which has the form P (t1 , . . . , tn ), where P is

an n-place predicate symbol of L and t1 , . . . , tn

are terms of L. If L contains equality (=), then

= is viewed as a two-place predicate. Consequently, if t1 and t2 are terms, then t1 = t2 is an

atomic formula.

atomic model

A model A in a language L

such that every n-tuple of elements of A satisfies a complete formula in T , the theory of

A. That is, for any a? ? An , there is an Lformula ? (x)

? such that A |= ? (a),

? and for any

L-formula

?,

either

T

?

x

?

?

(

x)

?

? ?(x)

? or

T ?x? ? (x)

? ? ¬?(x)

? . This is equivalent

to the complete type of every a? being principal.

Any finite model is atomic, as is the standard

model of number theory.

atom of a Boolean algebra

If (B, ?, ?,

?, 1, 0) is a Boolean algebra, a ? B is an atom

if it is a minimal element of B\{0}. For exam-

automorphism

ple, in the Boolean algebra of the power set of

any nonempty set, any singleton set is an atom.

considered to be an axiom of logic, not an axiom

of set theory.

automorphism

Let L be a first order language and let A be a structure for L. An automorphism of A is an isomorphism from A onto

itself. See isomorphism.

Axiom of Extensionality If two sets have the

same elements, then they are equal. This is one

of the axioms of Zermelo-Fraenkel set theory.

axiomatic set theory

A collection of statements concerning set theory which can be proved

from a collection of fundamental axioms. The

validity of the statements in the theory plays no

role; rather, one is only concerned with the fact

that they can be deduced from the axioms.

Axiom of Choice

Suppose that {X? }?? is

a 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-empty

sets

{X? }?? , there exists a function f : {X? }??

? ??

X? such that f (X? ) ? X? for each

? ? .

The existence of such a set Y or function f

can be proved from the Zermelo-Fraenkel axioms when there are only finitely many sets in

the family. However, when there are infinitely

many sets in the family it is impossible to prove

that such Y, f exist or do not exist. Therefore,

neither the Axiom of Choice nor its negation can

be proved from the axioms of Zermelo-Fraenkel

set theory.

Axiom of Comprehension

Also called Axiom of Separation. See Axiom of Separation.

Axiom of Constructibility Every set is constructible. See constructible set.

Axiom of Dependent Choice

of dependent choices.

Axiom of Infinity There exists an infinite set.

This is one of the axioms of Zermelo-Fraenkel

set theory. See infinite set.

Axiom of Regularity

Every non-empty set

has an ? -minimal element. More precisely, every non-empty set S contains an element x ? S

with the property that there is no element y ? S

such that y ? x. This is one of the axioms of

Zermelo-Fraenkel set theory.

Axiom of Replacement

If 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 of

Zermelo-Fraenkel set theory.

Axiom of Separation If P is a property and

X is a set, then there exists a set Y = {x ? X : x

satisfies 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, then

there exists a set Y = {X : X satisfies property

P }. Russell’s Paradox shows that the Axiom of

Comprehension is false for sets. See also Russell’s Paradox.

Axiom of Subsets

Same as the Axiom of

Separation. See Axiom of Separation.

See principle

Axiom of Determinancy

For any set X ?

?? , the game GX is determined. This axiom

contradicts the Axiom of Choice. See determined.

Axiom of Equality

If two sets are equal,

then they have the same elements. This is the

converse of the Axiom of Extensionality and is

© 2001 by CRC Press LLC

Axiom of Foundation

Same as the Axiom

of Regularity. See Axiom of Regularity.

Axiom of the Empty Set

? which has no elements.

There exists a set

Axiom of the Power Set

For every set X,

there exists a set P (X), the set of all subsets of

X. This is one of the axioms of Zermelo-Fraenkel set theory.

Axiom of the Unordered Pair If X and Y are

sets, then there exists a set {X, Y }. This axiom,

Axiom of Union

also known as the Axiom of Pairing, is one of

the axioms of Zermelo-Fraenkel set theory.

© 2001 by CRC Press LLC

Axiom of Union

For any set S, there exists

a set that is the union of all the elements of S.

base of number system

B

Baire class

The Baire classes B? are an increasing sequence of families of functions defined inductively for ? < ?1 . B0 is the set of

continuous functions. For ? > 0, f is in Baire

class ? if there is a sequence of functions {fn }

converging pointwise to f , with fn ? B?n and

?n < ? for each n. Thus, f is in Baire class

1 (or is Baire-1) if it is the pointwise limit of

a sequence of continuous functions. In some

cases, it is useful to define the classes so that if

f ? B? , then f ?

/ B? for any ? < ?. See also

Baire function.

Baire function

A function belonging to one

of the Baire classes, B? , for some ? < ?1 .

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 if

and only if f is Borel measurable, that is, if and

only if f ?1 (U ) is a Borel set for any open set

U.

Baire measurable function

A function f :

X ? Y , where X and Y are topological spaces,

such that the inverse image of any open set has

the Baire property. See Baire property. That is,

if V ? Y is open, then

f ?1 (V ) = U C = (U \ C) ? (C \ U ) ,

where U ? X is open and C ? X is meager.

Baire property

A set that can be written as

an open set modulo a first category or meager

set. That is, X has the Baire property if there is

an open set U and a meager set C with

X = U C = (U \ C) ? (C \ U ) .

Since the meager sets form a ? -ideal, this happens if and only if there is an open set U and

meager 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 LLC

Baire space

(1) A topological space X such

that no nonempty open set in X is meager (first

category). That is, no open set U = ? in X

may be written as a countable union of nowhere

dense sets. Equivalently, X is a Baire space if

and only if the intersection of any countable collection of dense open sets in X is dense, which is

true if and only if, for any countable collection of

closed sets {Cn } with empty interior, their union

?Cn also has empty interior. The Baire Category Theorem states that any complete metric

space 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 on

each copy of N. Thus, U is a basic open set in

NN if there is a finite sequence of natural numbers ? such that U is the set of all infinite sequences which begin with ? . The Baire space

is homeomorphic to the irrationals.

bar construction

For a group G, one can

construct a space BG as the geometric realization of the following simplicial complex. The

faces Fn in simplicial degree n are given by

(n + 1)-tuples of elements of G. The boundary

maps Fn ?? Fn?1 are given by the simplicial

boundary formula

n

(?1)i (g0 , . . . , g?i , . . . , gn )

i=0

where the notation g?i indicates that gi is omitted

to obtain an n-tuple. The ith degeneracy map

si : Fn ?? Fn+1 is given by inserting the group

identity element in the ith position.

Example: B(Z/2), the classifying space of

the group Z/2, is RP ? , real infinite projective

space (the union of RP n for all n positive integers).

The bar construction has many generalizations and is a useful means of constructing the

nerve of a category or the classifying space of a

group, which determines the vector bundles of

a manifold with the group acting on the fiber.

base of number system

The number b, in

use, when a real number r is written in the form

r=

N

j =??

rj b j ,

Bernays-G?del set theory

where each rj = 0, 1, ..., b ? 1, and r is represented in the notation

r = rN rN?1 · · · r0 .r?1 r?2 · · · .

For example, the base of the standard decimal

system 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 the

binary system; this is a “base 2” system. In

the base b system, the number 10215.2011 is

equivalent to the decimal number

1 ? 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 power

of the base b. See also radix.

Bernays-G?del set theory An axiomatic set

theory, which is based on axioms other than

those of Zermelo-Fraenkel set theory. BernaysG?del set theory considers two types of objects:

sets and classes. Every set is a class, but the

converse is not true; classes that are not sets

are called proper classes. This theory has the

Axioms of Infinity, Union, Power Set, Replacement, Regularity, and Unordered Pair for sets

from Zermelo-Fraenkel set theory. It also has

the following axioms, with classes written in :

(i.) Axiom of Extensionality (for classes):

Suppose that X and Y are two classes such that

U ? X if and only if U ? Y for all set U . Then

X = Y.

(ii.) If X ? Y, then X is a set.

(iii.) Axiom of Comprehension: For any formula F (X) having sets as variables there exists

a class Y consisting of all sets satisfying the formula F (X).

Bertrand’s postulate

If x is a real number

greater than 1, then there is at least one prime

number p so that x < p < 2x. Bertrand’s Postulate was conjectured to be true by the French

mathematician Joseph Louis Francois Bertrand

and later proved by the Russian mathematician

Pafnuty Lvovich Tchebychef.

Betti number

Suppose X is a space whose

homology groups are finitely generated. Then

the kth homology group is isomorphic to the direct sum of a torsion group Tk and a free Abelian

© 2001 by CRC Press LLC

group Bk . The kth Betti number bk (X) of X is

the rank of Bk . Equivalently, bk (X) is the dimension of Hk (X, Q), the kth homology group

with rational coefficients, viewed as a vector

space over the rationals. For example, b0 (X)

is the number of connected components of X.

bijection

A function f : X ? Y , between

two 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 an

x ? X such that f (x) = y).

See function.

binomial coefficient

(1) If n and k are nonnegative integers

with

k

?

n, then the binomial

n!

coefficient nk equals k!(n?k)!

.

(2) The binomial coefficient nk also represents the number of ways to choose k distinct

items from among n distinct items, without regard to the order of choosing.

(3)The binomial coefficient nk is the kth entry in the nth row of Pascal’s Triangle. It must be

noted that Pascal’s Triangle begins with row 0,

and each row begins with entry 0. See Pascal’s

triangle.

Binomial Theorem

If a and b are elements

of a commutative ring and

in

n is a non-negative

teger, then (a + b)n = nk=0 nk a k bn?k , where

n

k is the binomial coefficient. See binomial coefficient.

Bockstein operation In cohomology theory,

a cohomology operation is a natural transformation between two cohomology functors. If

0 ? A ? B ? C ? 0 is a short exact sequence of modules over a ring R, and if X ? Y

are topological spaces, then there is a long exact

sequence in cohomology:

· · · ? H q (X, Y ; A) ? H q (X, Y ; B) ?

H q (X, Y ; C) ?

H q+1 (X, Y ; A) ? H q+1 (X, Y ; B) ? . . . .

The homomorphism

? : H q (X, Y ; C) ? H q+1 (X, Y ; A)

is the Bockstein (cohomology) operation.

bounded quantifier

Bolzano-Weierstrass Theorem

Every

bounded sequence in R has a convergent subsequence. That is, if

{xn : n ? N} ? [a, b]

is an infinite sequence, then there is an increasing sequence {nk : k ? N} ? N such that

{xnk : k ? N} converges.

Boolean algebra

A non-empty set X, along

with two binary operations ? and ? (called union

and intersection, respectively), a unary operation (called complement), and two elements

0, 1 ? X which satisfy the following properties

for all A, B, C ? X.

(i.) A ? (B ? C) = (A ? B) ? C

(ii.) A ? (B ? C) = (A ? B) ? C

(iii.) A ? B = B ? A

(iv.) A ? B = B ? A

(v.) A ? (B ? C) = (A ? B) ? (A ? C)

(vi.) A ? (B ? C) = (A ? B) ? (A ? C)

(vii.) A ? 0 = A and A ? 1 = A

(viii.) There exists an element A so that A ?

A = 1 and A ? A = 0.

Borel measurable function

A function f :

X ? Y , for X, Y topological spaces, such that

the inverse image of any open set is a Borel set.

This is equivalent to requiring the inverse image

of any Borel set to be Borel. Any continuous

function is Borel measurable.

It is a theorem that f is Borel measurable

if and only if f is a Baire function. See Baire

function.

Borel set

The collection B of Borel sets of

a topological space X is the smallest ? -algebra

containing all open sets of X. That is, in addition

to containing open sets, B must be closed under

complements and countable intersections (and,

thus, is also closed under countable unions). For

comparison, the topology on X is closed under

arbitrary unions but only finite intersections.

Borel sets may also be defined inductively:

let 10 denote the collection of open sets and 01

the closed sets. Then for 1 < ? < ?1 , A ? ?0

if and only if

of B is in ?0 . Then the collection of all Borel

sets is

B = ??