First Concepts of Topology

First Concepts of Topology

(Parte 1 de 7)

published by The Mathematical Association of America

Editorial Committee

Ivan Niven, Chairman (1981-83) University of Oregon

Anneli Lax, Editor New York University

W. G. Chinn (1980-82) Basil Gordon (1 98&82) M. M. Sch8er (1979-81)

City College of San Francisco

University of Cdifornia, Los Angeles Stanford University

The Xew Mathematical Library (NML) was begun in 1961 by the

School Mathematics Study Group to make available to high school students short expository books on various topics not usually covered in the high school syllabus. In a decade the NML matured into a steadily growing series of some twenty titles of interest not only to the originally intended audience, but to college students and teachers at all levels. Previously published by Random House and L. W. Singer, the XML became a publication series of the Mathe- matical Association of America (MAA) in 1975. Under the auspices of the MAA the XML will continue to grow and will remain dedi- cated to its original and expanded purposes.

W. G. Chinn Sun Francisw Pubk Schok

and

N. E. Steenrod Princeton University

Illustrations by George H. Buehler

©Copyright 1966 by The Mathematical Association of America (Inc.)

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Published in Washington, D.C. by The Mathematical Association of America

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Print ISBN 978-0-88385-618-5

Electronic ISBN 978-0-88385-933-9 Manufactured in the United States of America

Note to the Reader

This book is one of a series written by professional mathematicians in order to make some important mathematical ideas interesting and under- standable to a large audience of high school students and laymen. Most of the volumes in the New Mathematical Libray cover topics not usually included in the high school curriculum; they vary in difficulty, and, even within a single book, some parts require a greater degree of concentration than others. Thus, while the reader needs little technical knowledge to understand most of these books, he will have to make an intellectual effort.

If the reader has so far encountered mathematics only in classroom work, he should keep in mind that a book on mathematics cannot be read

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The best way to learn mathematics is to do mathematics, and each book includes problems, some of which may require considerable thought. The reader is urged to acquire the habit of reading with paper and pencil in hand; in this way mathematics will become increasingly meaningful to him.

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The Contest Problem Book 1 Annual High School Mathematics Examinations 19661972. Compiled and with solutions by C. 7: Salkind and J. M. Earl Mathematical Methods in Science by George fdlya International Mathematical Olympiadsl95%1977. Compiled and with solutions by S. L. Greitzer The Mathematics of Games and Gambling by Edward K fuckel The Contest Problem Book IV Annual High School Mathematics Examinations 197S1982. Compiled and with solutions by R. A. Artino. A. M. Gaglione. and N. Shell The Role of Mathematics in Science by M. M. Schiffer and L. Bowden International Mathematical Olympiads I97w 985 and forty supplementary problems. Compiled and with solutions by Murray S. Klamkin Riddles of the Sphinx by Martin Gardner U.S.A. Mathematical Olympiads 1972-1986. Compiled and with solutions by Murray S. Klamkin Graphs and Their Uses by 0,vstein Ore. Revised and updated by Robin J. Wson Exploring Mathematics with Your Computer by Arthur Engel Game Theory and Strategy by Philip D. Strafln, JI: Episodes in Nineteenth and Twenthieth Century Euclidean Geometry by Ross

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Contents

Introduction

Part I Existence Theorems in Dimension 1

1. The first existence theorem 2. Sets and functions

3. Neighborhoods and continuity 4. Open sets and closed sets 5. The completeness of the real number system 6. Compactness 7. Connectedness 8. Topological properties and topological equivalences

9. A fixed point theorem 10. Mappings of a circle into a line

1. The pancake problems 12. Zeros of polynomials

Part I1 Existence Theorems in Dimension 2

13. Mappings of a plane into itself 14. The disk 15. Initial attempts to formulate the main theorem

16. Curves and closed curves 17. Intuitive definition of winding number 18. Statement of the main theorem 19. When is an argument not a proof? 20. The angle swept out by a curve 21. Partitioning a curve into short curves 2. The winding number W(p, y) 23. Properties of A ((p, y) and W(p, y) vii viii CONTENTS

24. Homotopies of curves 23. Constancy of the winding number 26. Proof of the main theorem 27. The circle winds once about each interior point 28. The fixed point property 29. Vector fields 30. The equivalence of vector fields and mappings 31. The index of a vector field around a closed curve 32. The mappings of a sphere into a plane 3. Dividing a ham sandwich 34. Vector fields tangent to a sphere 35. Complex numbers 36. Every polynomial has a zero 37. Epilogue: A brief glance at higher dimensional cases

Solutions for Exercises

Index 159

Introduction

Our purpose in writing this book is to show how topology arose, develop a few of its elements, and present some of its simpler applications.

Topology came to be recognized as a distinct area of mathematics about fifty years ago, and its major growth has taken place within the last thirty years. It is the most vigorous of the newer branches of mathe- matics and has been producing strong repercussions in most of the older branches. It got its start in response to the needs of analysis (the part of mathematics containing calculus itnd differential equations). However, topology is not a branch of anidysis. Instead, it is a kind of geometry. It is not an advanced form of geometry such as projective or differential geometry, but rather a primitive, rudimentary form-one which underlies all geometries. A striking fact about topology is that its ideas have pene- trated nearly all areas of mathematics. In most of these applications, topology supplies essential tools and concepts for proving certain basic propositions known as existence theorems.

Our presentation of the elements of topology will be centered around two existence theorems of analysis. The first, given in Part I, is funda- mental in the calculus and was known long before topology was recog- nized as a subject. In working out its proof, we shall develop basic ideas of topology. This will show how topology got started, and why it is useful. Our second main theorem, given in Part 1, is a generalization of the first from one to two dimensions. In contrast to the first, a. topological concept is needed for its formulation. Its proof exhibits that peculiar blending of numerical precision und rough qualitative geometry so char- acteristic of topology. Both theorems have numerous applications. We shall present those having the strongest topological flavor.

The beginnings of topology can be found in the work of Karl Weier- strass during the 1860’s in which he analyzed the concept of the limit of a function (as used in the calculus). In this endeavor, he reconstructed

2 FIRST CONCEPTS OF TOPOLOGY the real number system and revealed certain of its properties now called

“topological”. Then came Georg Cantor’s bold development of the theory of point sets (1874-1895); it provided a foundation on which topology eventually built its own house. A second aspect of topology, called com- binaforiul or algebraic topology, was initiated in the 1890’s by the re- markable work of Henri PoincarC dealing with the theory of integral calculus in higher dimensions. The first aspect, called set-theoretic topol- ogy, was placed on a firm foundation by F. Hausdorff and others during the period 1900-1910. A union of the combinatorial and set-theoretic aspects of topology was achieved first by L. E. J. Brouwer in his investiga- tion (1908-1912) of the concept of dimension. The unified theory was given a solid development in the period 1915-1930 by J. W. Alexander, P. L. Alexandrov, S. Lefschetz and others. Until 1930 topology was called anulysis situs. It was Lefschetz who first used and popularized the name fofiology by publishing a book with this title in 1930.

Since 1930 topology has been growing at an accelerated pace. To em- phasize this point we shall mention a few of topology’s achievements. It invaded the calculus of variations through the theory of critical points developed by M. Morse (Institute for Advanced Study, Princeton). It reinvigorated differential geometry through the work on fibre bundles by H. Whitney (Institute for Advanced Study, Princeton), the work on differential forms by G. de Rham (Lausanne), and the work on Lie groups by H. Hopf (Zurich). It enforced a minor revolution in modern

algebra through the development of new foundations for algebra and a new branch called homological algebra. Much of this work is due to S. Eilenberg (Columbia University) and S. MacLane (University of Chicago). Topology gave a new lease on life to algebraic geometry via the theory of sheaves and cohomology, and it has found important ap plications to partial differential equations through the works of J. Leray (Paris) and M. Atiyah (Oxford).

Applications of topology have been made to sciences other than mathe- matics, but nearly all of these occur through some intervening mathemati- cal subject. For example, the changes topology has made in differential geometry have initiated topological thinking in relativity theory. Topol- ogy has become a basic subject of mathematics, in fact, a necessity in many areas and a unifying force for nearly all of mathematics.

When a non-mathematician asks a topologist, “What is topology?”,

“What is it good for?”, the latter is at a disadvantage because the ques- tioner expects the kind of answer that can be given to analogous questions about trigonometry, namely, trigonometry deals with the determination of angles and is used to solve problems in surveying, navigation, and astronomy. The topologist cannot give such a direct answer; he can say, correctly, that topology is a kind of geometric thinking useful in many areas of advanced mathematics, but this does not satisfy the questioner who is after some of the flavor of the subject. The topologist can then bring out paper, scissors and paste, construct a Mobius band, and cut

INTRODUCTION 3 along its center line, or he can take some string and show how three separate loops can be enlaced without being linked. If he feels energetic, he can demonstrate how to take off his vest without removing his coat. These are parlor tricks, each based on a serious mathematical idea which would require at least several hours to explain. To present these tricks without adequate explanation is to present a caricature of topology.

To appreciate topology it is necessary to take the viewpoint of the mathematician and explore some of its successful applications. Most of these applications have in common that they occur in the proof of an existence theorem. An existence theorem is one which asserts that each of a certain broad class of problems has a solution of a particular kind. Such theorems are frequently the basic structure theorems of a subject. One of our principal aims is to demonstrate the power and flexibility of topology in proving existence theorems.

The existence theorem we shall prove in Part I answers the question:

When can an equation of the form f (x) = y be solved for x in terms of y ? Here f (x) stands for a function or formula (such as 2- 41 + x* ) defined for real numbers x in some interval [a, b] (such as [2, 41 ), and y denotes a real number (such as v). The problem is: Does there exist a number x in the interval [a, b] such that f (x) = y ? Formu- lated for the example it becomes: Is there an x between 2 and 4 such that 9- 4- = u? 3

We emphasize that we are not asking for methods of finding the value or values of x in special cases. Instead we are seeking a broad criterion, applicable to each of many different problems, to determine whether or not a solution exists. Once the criterion assures us that a particular prob- lem has a solution, we can start hunting for it with the knowledge that the search is not in vain.

The criterion given by our main theorem (stated in Section 1) requires the notion of the continuity of a function (defined in Section 3). The proof of the theorem (given in Sections 2-8) is based on two topological prop- erties of the interval [a, b] called compactness and connectedness. We give these concepts a thorough treatment because they are basic in modern mathematics.

The main theorem of Part I1 is an existence theorem which answers the question: When can a pair of simultaneous equations f(x, y) = a and g(x, y) = b be solved for x and y in terms of a and b ? A familiar example of such a problem is the pair of simultaneous linear equations x- 2y = 3 and 3x+y = 5; these can be solved readily by elimination. Here is a more difficult problem of the same type: Find a pair of numbers x, y satisfying the two equations

(Parte 1 de 7)

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