Table Of Content

Essentials of Topology with Applications TEXTBOOKS in MATHEMATICS Series Editor: Denny Gulick PUBLISHED TITLES COMPLEX VARIABLES: A PHYSICAL APPROACH WITH APPLICATIONS AND MATLAB® Steven G. Krantz ESSENTIALS OF TOPOLOGY WITH APPLICATIONS Steven G. Krantz INTRODUCTION TO ABSTRACT ALGEBRA Jonathan D. H. Smith INTRODUCTION TO MATHEMATICAL PROOFS: A TRANSITION Charles E. Roberts, Jr. LINEAR ALBEBRA: A FIRST COURSE WITH APPLICATIONS Larry E. Knop MATHEMATICAL AND EXPERIMENTAL MODELING OF PHYSICAL AND BIOLOGICAL PROCESSES H. T. Banks and H. T. Tran FORTHCOMING TITLES ENCOUNTERS WITH CHAOS AND FRACTALS Denny Gulick Essentials of Topology with Applications Steven G. Krantz Washington University St. Louis, Missouri, U.S.A. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20131121 International Standard Book Number-13: 978-1-4200-8975-2 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid- ity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or uti- lized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopy- ing, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To the memory of Paul Halmos. Table of Contents Preface xiii 1 Fundamentals 1 1.1 What Is Topology? . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 First Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 The Separation Axioms . . . . . . . . . . . . . . . . . . . . . . 12 1.5 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.6 Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.7 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.8 Path-Connectedness . . . . . . . . . . . . . . . . . . . . . . . 32 1.9 Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.10 Totally Disconnected Spaces . . . . . . . . . . . . . . . . . . . 36 1.11 The Cantor Set . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.12 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.13 Metrizability. . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.14 Baire’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.15 Lebesgue’s Lemma and Lebesgue Numbers . . . . . . . . . . . 52 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2 Advanced Properties of Topological Spaces 57 2.1 Basis and Sub-Basis . . . . . . . . . . . . . . . . . . . . . . . 57 2.2 Product Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.3 Relative Topology . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.4 First Countable, Second Countable, and So Forth . . . . . . . 62 vii viii 2.5 Compactifications . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.6 Quotient Topologies . . . . . . . . . . . . . . . . . . . . . . . . 68 2.7 Uniformities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.8 Morse Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.9 Proper Mappings . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.10 Paracompactness . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.11 An Application to Digital Imaging . . . . . . . . . . . . . . . 83 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3 Basic Algebraic Topology 95 3.1 Homotopy Theory . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.2 Homology Theory . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.2.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . 109 3.2.2 Singular Homology . . . . . . . . . . . . . . . . . . . . 110 3.2.3 Relation to Homotopy . . . . . . . . . . . . . . . . . . 120 3.3 Covering Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.4 The Concept of Index . . . . . . . . . . . . . . . . . . . . . . . 138 3.5 Mathematical Economics . . . . . . . . . . . . . . . . . . . . . 142 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4 Manifold Theory 157 4.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.2 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 5 Moore-Smith Convergence and Nets 173 5.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . 173 5.2 Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 6 Function Spaces 179 6.1 Preliminary Ideas . . . . . . . . . . . . . . . . . . . . . . . . . 179 6.2 The Topology of Pointwise Convergence . . . . . . . . . . . . 180 6.3 The Compact-Open Topology . . . . . . . . . . . . . . . . . . 181 ix 6.4 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . 182 6.5 Equicontinuity and the Ascoli-Arzela Theorem . . . . . . . . . 185 6.6 The Weierstrass Approximation Theorem . . . . . . . . . . . . 188 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 7 Knot Theory 197 7.1 What Is a Knot? . . . . . . . . . . . . . . . . . . . . . . . . . 197 7.2 The Alexander Polynomial . . . . . . . . . . . . . . . . . . . . 200 7.3 The Jones Polynomial . . . . . . . . . . . . . . . . . . . . . . 206 7.3.1 Knot Projections . . . . . . . . . . . . . . . . . . . . . 206 7.3.2 Reidemeister Moves . . . . . . . . . . . . . . . . . . . . 210 7.3.3 Bracket Polynomials . . . . . . . . . . . . . . . . . . . 214 7.3.4 Creation of a New Polynomial Invariant . . . . . . . . 216 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 8 Graph Theory 225 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 8.2 Fundamental Ideas of Graph Theory . . . . . . . . . . . . . . 227 8.3 Application to the K¨onigsberg Bridge Problem . . . . . . . . . 231 8.4 Coloring Problems . . . . . . . . . . . . . . . . . . . . . . . . 235 8.4.1 Modern Developments . . . . . . . . . . . . . . . . . . 240 8.4.2 Denouement . . . . . . . . . . . . . . . . . . . . . . . . 242 8.5 The Traveling Salesman Problem . . . . . . . . . . . . . . . . 242 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 9 Dynamical Systems 249 9.1 Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 9.1.1 Dynamical Systems . . . . . . . . . . . . . . . . . . . . 252 9.1.2 Stable and Unstable Fixed Points . . . . . . . . . . . . 255 9.1.3 Linear Dynamics in the Plane . . . . . . . . . . . . . . 257 9.2 Planar Autonomous Systems . . . . . . . . . . . . . . . . . . . 262 9.2.1 Ingredients of the Proof of Poincar´e-Bendixson . . . . . 263

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Essentials of Topology with Applications TEXTBOOKS in MATHEMATICS Series Editor: Denny Gulick PUBLISHED TITLES COMPLEX VARIABLES: A PHYSICAL APPROACH WITH APPLICATIONS AND MATLAB® Steven G. Krantz ESSENTIALS OF TOPOLOGY WITH APPLICATIONS Steven G. Krantz INTRODUCTION TO ABSTRACT ALGEBRA Jonathan D. H. Smith INTRODUCTION TO MATHEMATICAL PROOFS: A TRANSITION Charles E. Roberts, Jr. LINEAR ALBEBRA: A FIRST COURSE WITH APPLICATIONS Larry E. Knop MATHEMATICAL AND EXPERIMENTAL MODELING OF PHYSICAL AND BIOLOGICAL PROCESSES H. T. Banks and H. T. Tran FORTHCOMING TITLES ENCOUNTERS WITH CHAOS AND FRACTALS Denny Gulick Essentials of Topology with Applications Steven G. Krantz Washington University St. Louis, Missouri, U.S.A. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20131121 International Standard Book Number-13: 978-1-4200-8975-2 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid- ity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or uti- lized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopy- ing, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To the memory of Paul Halmos. Table of Contents Preface xiii 1 Fundamentals 1 1.1 What Is Topology? . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 First Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 The Separation Axioms . . . . . . . . . . . . . . . . . . . . . . 12 1.5 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.6 Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.7 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.8 Path-Connectedness . . . . . . . . . . . . . . . . . . . . . . . 32 1.9 Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.10 Totally Disconnected Spaces . . . . . . . . . . . . . . . . . . . 36 1.11 The Cantor Set . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.12 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.13 Metrizability. . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.14 Baire’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.15 Lebesgue’s Lemma and Lebesgue Numbers . . . . . . . . . . . 52 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2 Advanced Properties of Topological Spaces 57 2.1 Basis and Sub-Basis . . . . . . . . . . . . . . . . . . . . . . . 57 2.2 Product Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.3 Relative Topology . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.4 First Countable, Second Countable, and So Forth . . . . . . . 62 vii viii 2.5 Compactifications . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.6 Quotient Topologies . . . . . . . . . . . . . . . . . . . . . . . . 68 2.7 Uniformities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.8 Morse Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.9 Proper Mappings . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.10 Paracompactness . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.11 An Application to Digital Imaging . . . . . . . . . . . . . . . 83 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3 Basic Algebraic Topology 95 3.1 Homotopy Theory . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.2 Homology Theory . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.2.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . 109 3.2.2 Singular Homology . . . . . . . . . . . . . . . . . . . . 110 3.2.3 Relation to Homotopy . . . . . . . . . . . . . . . . . . 120 3.3 Covering Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.4 The Concept of Index . . . . . . . . . . . . . . . . . . . . . . . 138 3.5 Mathematical Economics . . . . . . . . . . . . . . . . . . . . . 142 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4 Manifold Theory 157 4.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.2 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 5 Moore-Smith Convergence and Nets 173 5.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . 173 5.2 Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 6 Function Spaces 179 6.1 Preliminary Ideas . . . . . . . . . . . . . . . . . . . . . . . . . 179 6.2 The Topology of Pointwise Convergence . . . . . . . . . . . . 180 6.3 The Compact-Open Topology . . . . . . . . . . . . . . . . . . 181 ix 6.4 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . 182 6.5 Equicontinuity and the Ascoli-Arzela Theorem . . . . . . . . . 185 6.6 The Weierstrass Approximation Theorem . . . . . . . . . . . . 188 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 7 Knot Theory 197 7.1 What Is a Knot? . . . . . . . . . . . . . . . . . . . . . . . . . 197 7.2 The Alexander Polynomial . . . . . . . . . . . . . . . . . . . . 200 7.3 The Jones Polynomial . . . . . . . . . . . . . . . . . . . . . . 206 7.3.1 Knot Projections . . . . . . . . . . . . . . . . . . . . . 206 7.3.2 Reidemeister Moves . . . . . . . . . . . . . . . . . . . . 210 7.3.3 Bracket Polynomials . . . . . . . . . . . . . . . . . . . 214 7.3.4 Creation of a New Polynomial Invariant . . . . . . . . 216 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 8 Graph Theory 225 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 8.2 Fundamental Ideas of Graph Theory . . . . . . . . . . . . . . 227 8.3 Application to the K¨onigsberg Bridge Problem . . . . . . . . . 231 8.4 Coloring Problems . . . . . . . . . . . . . . . . . . . . . . . . 235 8.4.1 Modern Developments . . . . . . . . . . . . . . . . . . 240 8.4.2 Denouement . . . . . . . . . . . . . . . . . . . . . . . . 242 8.5 The Traveling Salesman Problem . . . . . . . . . . . . . . . . 242 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 9 Dynamical Systems 249 9.1 Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 9.1.1 Dynamical Systems . . . . . . . . . . . . . . . . . . . . 252 9.1.2 Stable and Unstable Fixed Points . . . . . . . . . . . . 255 9.1.3 Linear Dynamics in the Plane . . . . . . . . . . . . . . 257 9.2 Planar Autonomous Systems . . . . . . . . . . . . . . . . . . . 262 9.2.1 Ingredients of the Proof of Poincar´e-Bendixson . . . . . 263

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