Continuous Transformations in Analysis: With an Introduction to Algebraic Topology

448 Pages·1955·5.936 MB·English

by  T. Rado, P. V. Reichelderfer (auth.)

DIE GRUNDLEHREN-DER
MATHEMATISCHEN
WISSENSCHAFTEN
IN EINZELDARSTELLUNGEN MIT BESONDERER
BERUCKSICHTIGUNG DER ANWENDUNGSGEBIETE
HERAUSGEGEBEN VON
R. GRAMMEL .-E. BOPF . H. HOPF . F. RELLICH
F. K. SCHMIDT· B. L. VAN DER W AERDEN
BAND LXXV
CONTINUOUS TRANSFORMATIONS
IN ANALYSIS
WITH AN INTRODUCTION TO
ALGEBRAIC TOPOLOGY
BY
T.RADO
AND
P. V. REICHELDERFER
SPRINGER-VERLAG
BERLIN· GOTTINGEN . HEIDELBERG
1955
CONTINUOUS
TRANSFORMATIONS IN
ANALYSIS
WITH AN INTRODUCTION TO
ALGEBRAIC TOPOLOGY
BY
T. RADO
RESEARCH PROFESSOR OF MATHEMATICS
IN THE OHIO STATE UNIVERSITY
AND
P. V. REICHELDERFER
PROFESSOR OF MATHEMATICS IN THE OHIO STATE UNIVERSITY
MIT 53 TEXTABBILDUNGEN
SPRINGER-VERLAG
BERLIN· GOTTINGEN . HEIDELBERG
1955
ALLE RECHTE,
INSBESONDERE DAS DER DBERSETZUNG IN FREMDE SPRACHEN,
VORBEHALTEN
OHNE AUSDRDcKLICHE GENEHMIGUNG DES VERLAGES
1ST ES AUCH NICHT GESTATTET, DIESES BUCH ODER TEILE DARAUS
AUF PHOTOMECHANISCHEM WEGE (PHOTOKOPIE, MIKROKOPIE) ZU VERVIELFALTIGEN
ISBN-13 978-3-642-85991-5 e-ISBN-13 978-3-642-85989-2
001: 10.1007/978-3-642-85989-2
Softcover reprint of the hardcover 1st edition 1955
COPYRIGHT 1955
BY SPRINGER-VERLAG OHG. IN BERLIN, GOTTINGEN AND HEIDELBERG
Introduction.
The general objective of this treatise is to give a systematic presenta
tion of some of the topological and measure-theoretical foundations
of the theory of real-valued functions of several real variables, with
particular emphasis upon a line of thought initiated by BANACH, GEOCZE,
LEBESGUE, TONELLI, and VITALI. To indicate a basic feature in this
line of thought, let us consider a real-valued continuous function I(u)
of the single real variable tt. Such a function may be thought of as
defining a continuous translormation T under which x = 1( u) is the
image of u. About thirty years ago, BANACH and VITALI observed that
the fundamental concepts of bounded variation, absolute continuity,
and derivative admit of fruitful geometrical descriptions in terms of
the transformation T: x = 1( u) associated with the function 1( u). They
further noticed that these geometrical descriptions remain meaningful
for a continuous transformation T in Euclidean n-space Rff, where T
is given by a system of equations of the form
1-/(1 ff)
X-I U, ... ,tt ,.",
and n is an arbitrary positive integer. Accordingly, these geometrical
descriptions can be used to define, for continuous transformations in
Euclidean n-space Rff, n-dimensional concepts 01 bounded variation and
absolute continuity, and to introduce a generalized Jacobian without
reference to partial derivatives. These ideas were further developed,
generalized, and modified by many mathematicians, and significant
applications were made in Calculus of Variations and related fields
along the lines initiated by GEOCZE, LEBESGUE, and TONELLI. In
their fundamental aspects, these researches constitute a study of
continuous transformations in Euclidean n-space Rn in terms of Point
Set Theory, Algebraic Topology, and the modern Theory of Functions
of Real Variables. The purpose of this volume is to give a systematic
account of some of the basic concepts, methods, and results in this
general area.
The development of this program involves a study of several in
dividual topics which may deserve mention here. The n-dimensional
concepts of bounded variation, absolute continuity and generalized
Jacobians, already referred to above, are discussed thoroughly from
various points of view in Part IV. The topological index, a concept of
VI Introduction.
general utility in Mathematics, is studied in detail in §§ 11.2, 11.3 for the
case of Euclidean n-space, and in §§ VI.1, VI.2 the important case of
the plane is given special consideration. The transformation of multiple
integrals, a troublesome issue even under the very restrictive assumptions
customary in classical areas of Analysis, is discussed in several forms and
in extreme generality in various sections of Parts IV, V, VI. Local linear
approximations in Euclidean n-space are discussed in § V.2 in terms of the
various types of total differentials. Since this treatise has been designed
for use by mathematicians primarily interested in Analysis, it did not
seem appropriate to assume that the reader is familiar with Algebraic
Topology to the extent necessary for our purposes. Accordingly, a
self-contained exposition of the required background material in this
field is given in §§ 1.4, 1.5, 1.6, 1.7, 11.1. This exposition covers, in a
very detailed manner, the basic algebraic apparatus needed in general
cohomology theory, as well as some of the fundamental topics in the
topology of Euclidean n-space R"'. Thus this exposition may serve
as a first introduction to Algebraic Topology, especially since a definite
effort has been made to prepare the reader for the study of the excellent
comprehensive treatises in this field by ALEXANDROFF-HoPF, LEFSCHETZ,
ElLENBERG-STEENROD (see the Bibliography).
While our main concern is the discussion of a general theory appli
cable in Euclidean n-space, we included a detailed study of the special
cases n = 1 and n = 2. The case n = 2 is considered in Part VI, with
due emphasis upon the special features which are present in this case
and are lost in the general case n> 2. The case n = 1, discussed in § V.1,
furnishes a concrete picture of the general abstract concepts involved,
and the study of this case yields instructive geometrical interpretations
for many of the basic ideas and theorems in the modem theory of
real-valued functions of a single real variable.
The method of presentation in this volume is based upon the assump
tion that the reader has progressed beyond the stage of basic training
in the theory of Functions of Real Variables, Point Set Theory, and
Group Theory. The background material in these fields is merely
summarized in §§ 1.1, 1.2, 1.3, 111.1. For the convenience of the reader,
an Index of the terms and symbols used in the book has been added.
The Bibliography contains only a list of treatises directly related to
our subject. References to literature concerning individual topics are
given in footnotes throughout the text, with no attempt at completeness.
The drawings for the diagrams were prepared by our colleague
F. W. NIEDENFUHR. We take pleasure in thanking him for his help.
Columbus, Ohio. T.RADO
January, 1955. P. V. REICHELDERFER.
Table of Contents.
Introduction.
Part I. Background in Topology.
§ 1. 1. Survey of general topology 1
§ 1.2. Survey of Euclidean spaces 19
§ 1.3. Survey of Abelian groups. 26
§ 104. MAYER complexes 30
§ 1.5. Formal complexes . . . . 45
§ 1.6. General cohomology theory 63
§ I. 7. Cohomology groups in Euclidean spaces 98
nn.
Part II. Topological study of continuous transformations in
§ ILL Orientation in Rn . . . . . . . . . . . 110
§ 11.2. The topological index . . . . . . . . . . . . . 120
§ II.3. Multiplicity functions and index functions . . . . 145
Part III. Background in Analysis.
§ IIL1. Survey of functions of real variables. . . . . . 190
§ III.2. Functions of open intervals in Rn . . . . . . . 201
nn.
Part IV. Bounded variation and absolute continuity in
§ IV.1. Measurability questions . . . . . . . . . . . . . . . . . 212
§ IV.2. Bounded variation and absolute continuity with respect to a base-
function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
§ IV.3. Bounded variation and absolute continuity with respect to a multi-
plicity function . . . . . . . . . . . . . . . . . . . . . . . .. 232
§ IVA. Essential bounded variation and absolute continuity . . . 249
§ IV.5. Bounded variation and absolute continuity in the BANACH sense 277
nn.
Part V. Differentiable transformations in
§ V.1. Continuous transformations in Rl ......... . 292
§ V.2. Local approximations in Rn . . . . . . . . . . . . 319
§ V.3. Special classes of differentiable transformations in Rn. 349
Part VI. Continuous transformations in n2
§ VL1. The topological index in R2 . . . . . . . . . . . . 377
§ VI.2. Special features of continuous transformations in R2 . 402
§ VL3. Special classes of differentiable transformations in R2 415
Index .... 439
Bibliography. . . . . . 442
Part 1. Background in topology.
§ 1.1. Survey of general topology.
1.1.1. Sets and mappings. If X is a set, then x E X means that x
is an element of X, while y Ef X means that y is not an element of X.
A set which has a single element x is denoted by (x). On logical grounds,
it is necessary to distinguish between an object x and the set (x) con
sisting of the single object x. However, as a matter of notational con
venience we shall use frequently the same symbol for an object x and
the set consisting of the single object x. It is also convenient to use
the concept of the empty set which has no element. The empty set
will be denoted by the symbol 0.
A set may be finite or infinite. A set X is termed countable if there
exists a one-to-one correspondence between X and a set consisting of
some or all of the positive integers 1, 2, ... , or if X is the empty set.
Thus the elements of a non-empty countable set X can be arranged into
a (finite or infinite) sequence Xl' x2, ...•
If S (x) designates a certain statement relating to the object x, then
we write
E = {xIS(x)}
to state that E is the set of those objects x for which the statement
S (x) holds. For example, if A and B are two given sets, then their
union A U B consists of those objects x which belong to at least one of
n
the sets A and B, and their intersection A B consists of those objects x
which belong to both A and B. These definitions may be stated in the
form of the following equations.
A UB = {xlxEA or xEB},
n
A B = {x I x E A and x E B} .
The concepts of union and intersection apply to any number of sets.
If F is a family of sets, then
U X = {x I x E X for some X E F} ,
XEF
n X = {x I x E X for every X E F}.
Xc:F •
An
Two sets A, B are said to be disjoint if B = 0.
Rado and Reichelderfer, Continuous Transformation~.
2 Part 1. Background in topology.
If every element of a set A belongs to a set B, then A is termed a
subset of B, and we write A(B or equivalently B)A. If A is not a
subset of B, then we write A<j:B. As a matter of convenience, the empty
set is considered a subset of every set A.
If A is a subset of X, then X and A are said to constitute a pair
(X, A). If X is any set, then (X, 0) is a pair. If (X, A) and (Y, B)
are two pairs such that X(Y and A(B, then we write (X, A) ((Y, B).
If S (X, then the complement CxS of S with respect to X is the
set of those elements of X which do not belong to S. In symbols
CxS={xlxEX, xEfS}.
If S is also a subset of some other set Y, then Cy 5 and CxS are dif
ferent sets. If we consider, in a certain situation, only subsets S of
a fixed set X, then we write C S instead of Cx S as a matter of con
vemence.
The difference A - B of two sets A and B is defined by the formula
A - B = {xlxEA, xEfB}.
Consider a fixed set X and its subsets (thus C will stand for comple
ment with respect to X). The symbols U, n, c, -, introduced above,
give rise to a number of identities involving subsets of X. Some examples
follow.
CCA =A, A-B=AnCB.
AUA =A, AnA =A, AUB=BUA, AnB=BnA.
AU (B U C) = (A U B) U C, An (B n C) = (A n B) n C.
An (B UC) = (A n B) U (A n C), AU (Bn C) = (A U B) n (A U C).
C(A UB) = CAnCB, C(AnB) = CA UCB.
Similar identities hold in situations involving an arbitrary number of
subsets of X. For example, if F is any family of subsets of X, then
CU5 = ncs, SE F,
cns = UCS, SE F.
Furthermore, there exist certain logical relationships between the sym
bols already introduced. For example, A ( B if and only if A U B = B.
Similarly, A (B if and only if An B = A.
If S is a subset of X, then the function c(x, 5) defined by the formula
1 if xES,
{
c(x, 5) = o 'f E C5 ,
1 X
§ 1.1. Survey of general topology. 3
is termed the characteristic lunction of 5 in X. For any two subsets
A, B of X we have the identity
c (x, A UB) + c (x, A n B) = c (x, A) + c (x, B).
If Xl' X are two sets, then their Cartesian product Xl X X is the
2 2
set of all ordered pairs (Xl' x2) such that xlEXl, x2EX2• For example,
if (X,A) is a pair of sets, and I:O:;;'u;:;;;;1 is the unit interval on the real
number-line, then one has the Cartesian products XxI, A xl, Xx u,
A x u (where u is any element of I), and the pairs
(XxI, A xl), (Xxu, A xu),
as well as the inclusion
(Xxu, A xu) «(XxI, A xl,.
Given two sets X and Y, a mapping (or translormation) I:X -+ Y
from X into Y is a rule which assigns to each xE X a unique element
y = I xE Y as its image.
Given a mapping I: X -+ Y and a subset A =1= 0 of X, we denote by
II
A the mapping from A into Y which agrees with I on A. Thus
(f IA ) x = I x for x EA.
I
The mapping I A is referred to as I cut down to A or also as the restriction
of I to A.
Given a mapping I:X -~ Y and a subset A of X, the image I A of
A under I is the set of the images I x of the elements of A. If B is a
subset of Y, then 1-1 B is the set of those elements x of X for which
I x E B. In particular, if Yo is an element of Y, then 1-1 Yo is the set of
those elements x of X for which I x = Yo'
Given a mapping I: X -+ Y, I is said to be onto if I X = Y, and I is
said to be one-to-one if Xl =1= X2 implies that I Xl =1= I x2• If I is both onto
and one-to-one, then I is termed bi-unique. If I is bi-unique, then for
each yE Y the set I-ly consists of a single point of X. Thus in this
case we have at our disposal the mapping 1-1: Y -+X, the inverse of I.
The following set of identities, relating to an arbitrary mapping
I:X -+ Y, will be constantly used. If F is any family of subsets of X,
then
IUA=UIA, AEF,
InA(nIA, AEF.
If AI' A 2 are subsets of X such that Al ( A 2' then I Al (j A 2' If ([> is
any family of subsets of Y, then
1-1 U B = U 1-1 B , 1-1 n B = n 1-1 B , B E ([> •
1*
4 Part 1. Background in topology.
If B1, B2 are any two subsets of Y, then
1-1 (B2 - B1) = 1-1 B2 - 1-1 B1.
If A, B are subsets of X and Y respectively, then
Consider now two mappings I:X~Y, g:Y~Z. Then the product
mapping g I: X ~ Z is defined by the formula
(g f) x = g (f x), x EX.
Note that a product of mappings is to be read Irom the right to the lelt.
Given a mapping I: X ~ Y, consider a subset A of X and a subset
B of Y. If I A (B, then I is said to be a mapping from the pair (X, A)
into the pair (Y, B), and one writes
I:
(X, A) ~ (Y, B).
If I is a mapping from (X, A) into (Y, B) and g is a mapping from (Y, B)
into (Z, C) then clearly gl is a mapping from (X, A) into (Z, C).
If X is any set, then the identity mapping i:X ~X on X is defined
by ix= x for x EX. If A is a subset of X, then the mapping iIA:A ~X
is termed the inclusion mapPing from A into X.
Consider now two pairs (X, A), (Y, B) such that (X, A) ((Y, B),
and let j be the identity mapping on Y. Then the mapping
I
j X: (X, A) ~ (Y, B)
is termed the inclusion mapping from (X, A) into (Y, B).
If one has to consider several pairs of sets and certain mappings
from these pairs, it is helpful to set up a mapping diagram. We give
two examples.
Let (X, A) be a pair of sets, and let I:O:;;;'u:;;;'1 be the unit interval
on the real number-line. The following diagram will be referred to later
on as the lirst homotopy
diagram.
In this diagram, s
and t denote two real
numbers in I. The
mappings is, it are the
inclusion mappings cor
responding to the In
Fig. 1. clusions
(Xxs, A xs) ((XxI, A xl), (Xxt, A xt) ((XxI, A xl).