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111111111| | | | | | | l l l > I I M INTRODUCTORY TOPOLOGY Exercises and Solutions S e c o n d Edition IN T R O D U C T O R Y TOPOLOGY Exercises and Solutions Second Edition This page intentionally left blank INTRODUCTORY TOPOLOGY Exercises and Solutions Second Edition Mohammed Hichem Mortad University of Oran 1, Ahmed Ben Bella, Algeria World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI Published by World Scientific Publishing Co Pte Ltd 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library o f Congress Cataloging-in-Publication Data Names: Mortad, Mohammed Hichem, 1978- Title: Introductory topology : exercises and solutions / by Mohammed Hichem Mortad (University o f Oran, Algeria) Description: 2nd edition |New Jersey : World Scientific, 2016 | Includes bibliographical references and index Identifiers: LCCN 2016030117 |ISBN 9789813146938 (hardcover : alk paper) | ISBN 9789813148024 (pbk : alk paper) Subjects: LCSH: Topology—Problems, exercises, etc Classification: LCC QA611 M677 2016 |DDC 514.076«dc23 LC record available at https://lccn.loc.gov/2016030117 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Copyright © 2017 by World Scientific Publishing Co Pte Ltd A ll rights reserved This book, or parts thereof, may not be reproduced in anyform or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher For photocopying o f material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, M A 01923, USA In this case permission to photocopy is not required from the publisher Printed in Singapore Contents Preface ix Preface to the Second Edition xiii Notation and Terminology XV 0.1 Notation XV 0.2 Terminology xvi Part 1 Exercises 1 Chapter 1 General Notions: Sets, Functions et al 3 1 1 Essential Background 3 1 1 1 Sets 3 1.1.2 Functions 9 1.1.3 Equivalence Relation 15 1.1.4 Consequences of The Least Upper Bound Property 15 1.1.5 Countability 17 1.2 Exercises With Solutions 19 1.3 More Exercises 24 Chapter 2 Metric Spaces 27 2.1 Essential Background 27 2.1.1 Definitions and Examples 27 2.1.2 Important Sets in Metric Spaces 30 2.1.3 Continuity in Metric Spaces 32 2.1.4 Equivalent Metrics 34 2.2 True or False: Questions 36 2.3 Exercises With Solutions 36 2.4 Tests 42 2.5 More Exercises 43 Chapter 3 Topological Spaces 47 3.1 Essential Background 47 3.1.1 General Notions 47 3.1.2 Separation Axioms 49 3.1.3 Closures, Interiors, Limits Points, et al 50 v VI CONTENTS 3.1.4 Bases and Subbases 56 3.1.5 The Subspace Topology 58 3.1.6 The Product and Quotient Topologies 59 3.2 True or False: Questions 61 3.3 Exercises With Solutions 63 3.4 Tests 72 3.5 More Exercises 73 Chapter 4 Continuity and Convergence 77 4.1 Essential Background 77 4.1.1 Continuity 77 4.1.2 Convergence 82 4.1.3 Sequential Continuity 83 4.1.4 A Word on Infinite Product Topology 84 4.2 True or False: Questions 86 4.3 Exercises With Solutions 87 4.4 Tests 94 4.5 More Exercises 95 Chapter 5 Compact Spaces 99 5.1 Essential Background 99 5.1.1 Compactness: General Notions 99 5.1.2 Compactness and Continuity 102 5.1.3 Sequential Compactness and Total Boundedness 104 5.2 True or False: Questions 106 5.3 Exercises With Solutions 107 5.4 Tests 112 5.5 More Exercises 113 Chapter 6 Connected Spaces 117 6.1 Essential Background 117 6.1.1 Connectedness 117 6.1.2 Components 119 6.1.3 Path-connectedness 120 6.2 True or False: Questions 122 6.3 Exercises With Solutions 123 6.4 Tests 125 6.5 More Exercises 126 Chapter 7 Complete Metric Spaces 129 7.1 Essential Background 129 7.1.1 Completeness 129 7.1.2 Fixed Point Theorem 135 CONTENTS vii 7.1.3 A Word on Integral Equations 137 7.2 True or False: Questions 140 7.3 Exercises With Solutions 141 7.4 Tests 148 7.5 More Exercises 148 Chapter 8 Function Spaces 153 8.1 Essential Background 153 8.1 1 Types of Convergence 153 8.1 2 Weierstrass Approximation Theorem 155 8.1.3 The Arzela-Ascoli Theorem 155 8.2 True or False: Questions 157 8.3 Exercises With Solutions 157 8.4 Tests 159 8.5 More Exercises 159 Part 2 Solutions 161 Chapter 1 General Notions: Sets,Functions et al 163 1 2 Solutions to Exercises 163 Chapter 2 Metric Spaces 181 2.2 True or False: Answers 181 2.3 Solutions to Exercises 183 2.4 Hints/Answers to Tests 201 Chapter 3 Topological Spaces 203 3.2 True or False: Answers 203 3.3 Solutions to Exercises 209 3.4 Hints/Answers to Tests 241 Chapter 4 Continuity and Convergence 243 4.2 True or False: Answers 243 4.3 Solutions to Exercises 247 4.4 Hints/Answers to Tests 266 Chapter 5 Compact Spaces 269 5.2 True or False: Answers 269 5.3 Solutions to Exercises 272 5.4 Hints/Answers to Tests 293 Chapter 6 Connected Spaces 295 6.2 True or False: Answers 295 6.3 Solutions to Exercises 298 6.4 Hints/Answers to Tests 307 viii CONTENTS Chapter 7 C o m p le te M e tr ic Spaces 309 7.2 True or False: Answers 309 7.3 Solutions to Exercises 313 7.4 Hints/Answers to Tests 342 Chapter 8 F u n ction Spaces 343 8.2 True or False: Answers 343 8.3 Solutions to Exercises 345 8.4 Hints/Answers to Tests 350 Bibliography 351 Index 353 Preface Topology is a major area in mathematics At an undergraduate level, at a research level, and in many areas of mathematics (and even outside of mathematics in some cases), a good understanding of the basics of the theory of general topology is required Many students find the course nTopologyn (at least in the beginning) a bit confusing and not too easy (even hard for some of them) to assimilate It is like moving to a different place where the habits are not as they used to be, but in the end we know that we have to live there and get used to it They are usually quite familiar with the real line R and its properties So when they study topology they start to realize that not everything true in M needs to remain true in an arbitrary topological space For in stance, there are convergent sequences which have more than one limit, the identity mapping is not always continuous, a normally convergent series need not converge (although the latter is not within the scope of this book) So in many references, they use the word "usual topology of R ", a topology in which things are as usual! while there are many other "unusual topologies" where things are not so "usual"! The present book offers a good introduction to basic general topol ogy throughout solved exercises and one of the main aims is to make the understanding of topology an easy task to students by proposing many different and interesting exercises with very detailed solutions, something that it is not easy to find in another manuscript on the same subject in the existing literature Nevertheless, and in order that this books gives its fruits, we do advise the reader (mainly the students) to use the book in a clever way inasmuch as while the best way to learn mathematics is by doing exercises, the worst way of doing exercises is to read the solution without thinking about how to solve the exercises (at least for some time) Accordingly, we strongly recommend the student to attempt the exercises before consulting the solutions As a Chinese proverb says: If you give someone a fish, then you have given him to eat for one day, but if you teach him how to fish, then you have given him food for everyday So we hope the students are going to learn to "fish" using this book IX