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  • TOPOLOGY WITHOUT TEARS1

    SIDNEY A. MORRIS

    Version of July 31, 20052

    1 cCopyright 1985-2005. No part of this book may be reproduced by any process without prior writtenpermission from the author. If you would like a printable version of this book please e-mail your name,address, and commitment to respect the copyright (by not providing the password, hard copy or soft copyto anyone else) to [email protected]

    2This book is being progressively updated and expanded; it is anticipated that there will be about fifteenchapters in all. Only those chapters which appear in colour have been updated so far. If you discover anyerrors or you have suggested improvements please e-mail: [email protected]

  • Contents

    Title Page and Copyright Statement 1

    Contents 2

    Introduction 4

    Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

    Readers Locations and Professions . . . . . . . . . . . . . . . . . . . . . . . . . 5

    Readers Compliments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    1 Topological Spaces 9

    1.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

    1.2 Open Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

    1.3 Finite-Closed Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

    1.4 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

    2 The Euclidean Topology 31

    2.1 Euclidean Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

    2.2 Basis for a Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

    2.3 Basis for a Given Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

    2.4 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

    3 Limit Points 52

    3.1 Limit Points and Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

    3.2 Neighbourhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

    3.3 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

    3.4 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

    4 Homeomorphisms 66

    4.1 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

    4.2 Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

    2

  • CONTENTS 3

    4.3 Non-Homeomorphic Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

    4.4 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

    5 Continuous Mappings 86

    5.1 Continuous Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

    5.2 Intermediate Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

    5.3 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

    6 Metric Spaces 100

    6.1 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

    6.2 Convergence of Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

    6.3 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

    6.4 Contraction Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

    6.5 Baire Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

    6.6 Hausdorff Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

    6.7 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

    7 Compactness 157

    7.1 Compact Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

    7.2 The Heine-Borel Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

    7.3 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

    8 Finite Products 170

    8.1 The Product Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

    8.2 Projections onto Factors of a Product . . . . . . . . . . . . . . . . . . . . . . . . 175

    8.3 Tychonoffs Theorem for Finite Products . . . . . . . . . . . . . . . . . . . . . . . 180

    8.4 Products and Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

    8.5 Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

    8.6 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

    Appendix 1: Infinite Sets 190

    Appendix 2: Topology Personalities 213

    Bibliography 220

    Index 234

  • Introduction

    Topology is an important and interesting area of mathematics, the study of which will not only

    introduce you to new concepts and theorems but also put into context old ones like continuous

    functions. However, to say just this is to understate the significance of topology. It is so

    fundamental that its influence is evident in almost every other branch of mathematics. This

    makes the study of topology relevant to all who aspire to be mathematicians whether their first

    love is (or will be) algebra, analysis, category theory, chaos, continuum mechanics, dynamics,

    geometry, industrial mathematics, mathematical biology, mathematical economics, mathematical

    finance, mathematical modelling, mathematical physics, mathematics of communication, number

    theory, numerical mathematics, operations research or statistics. (The substantial bibliography

    at the end of this book suffices to indicate that topology does indeed have relevance to all these

    areas, and more.) Topological notions like compactness, connectedness and denseness are as

    basic to mathematicians of today as sets and functions were to those of last century.

    Topology has several different branches general topology (also known as point-set topology),

    algebraic topology, differential topology and topological algebra the first, general topology,

    being the door to the study of the others. I aim in this book to provide a thorough grounding in

    general topology. Anyone who conscientiously studies about the first ten chapters and solves at

    least half of the exercises will certainly have such a grounding.

    For the reader who has not previously studied an axiomatic branch of mathematics such as

    abstract algebra, learning to write proofs will be a hurdle. To assist you to learn how to write

    proofs, quite often in the early chapters, I include an aside which does not form part of the proof

    but outlines the thought process which led to the proof.

    4

  • CONTENTS 5

    Asides are indicated in the following manner:

    In order to arrive at the proof, I went through this thought process, which might well

    be called the discovery or experiment phase.

    However, the reader will learn that while discovery or experimentation is often

    essential, nothing can replace a formal proof.

    There are many exercises in this book. Only by working through a good number of exercises

    will you master this course. I have not provided answers to the exercises, and I have no intention

    of doing so. It is my opinion that there are enough worked examples and proofs within the text

    itself, that it is not necessary to provide answers to exercises indeed it is probably undesirable

    to do so. Very often I include new concepts in the exercises; the concepts which I consider most

    important will generally be introduced again in the text.

    Harder exercises are indicated by an *.

    Finally, I should mention that mathematical advances are best understood when considered

    in their historical context. This book currently fails to address the historical context sufficiently.

    For the present I have had to content myself with notes on topology personalities in Appendix

    2 - these notes largely being extracted from Mac [137]. The reader is encouraged to visit the

    website Mac [137] and to read the full articles as well as articles on other key personalities. But

    a good understanding of history is rarely obtained by reading from just one source.

    In the context of history, all I will say here is that much of the topology described in this book

    was discovered in the first half of the twentieth century. And one could well say that the centre

    of gravity for this period of discovery is, or was, Poland. (Borders have moved considerably.) It

    would be fair to say that World War II permanently changed the centre of gravity. The reader

    should consult Appendix 2 to understand this cryptic comment.

    Acknowledgment

    Portions of earlier versions of this book were used at LaTrobe University, University of New

    England, University of Wollongong, University of Queensland, University of South Australia, City

    College of New York, and the University of Ballarat over the last 30 years. I wish to thank those

    students who criticized the earlier versions and identified errors. Special thanks go to Deborah

  • 6 CONTENTS

    King and Allison Plant for pointing out numerous errors and weaknesses in the presentation.

    Thanks also go to several other colleagues including Carolyn McPhail, Ralph Kopperman, Karl

    Heinrich Hofmann, Rodney Nillsen, Peter Pleasants, Geoffrey Prince, Bevan Thompson and Ewan

    Barker who read earlier versions and o

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