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12: Metric Space Topology: Examples, Exercises and Solutionsby W-S CheungVol.. Title: Metric space topology : examples, exercises and solutions / Wing-Sum Cheung, the University of Hong
Series on Analysis, Applications and Computation – Vol 12 ISAAC Series on Analysis, Applications and Computation ISSN: 1793-4702 Series Editors: Heinrich G W Begehr (Freie Univ Berlin, Germany) Robert Pertsch Gilbert (Univ Delaware, USA) Tao Qian (Univ of Macau, China) M W Wong (York Univ., Canada) Advisory Board Members: Mikhail S Agranovich (Moscow Inst of Elec & Math., Ryuichi Ashino (Osaka Kyoiku Univ., Japan), Russia), Alain Bourgeat (Univ de Lyon, France), Victor Burenkov (Cardiff Univ., UK), Jinyuan Du (Wuhan Univ., China), Antonio Fasano (Univ di Firenez, Italy), Massimo Lanza de Cristoforis (Univ di Padova, Italy), Bert-Wolfgang Schulze (Univ Potsdam, Germany), Masahiro Yamamoto (Univ of Tokyo, Japan) & Armand Wirgin (CNRS-Marseille, France) Published Vol 12: Metric Space Topology: Examples, Exercises and Solutions by W-S Cheung Vol 11: Hardy Operators on Euclidean Spaces and Related Topics by S Lu, Z Fu, F Zhao & S Shi Vol 10: Fractional Differential Equations and Inclusions: Classical and Advanced Topics by S Abbas, M Benchohra, J E Lazreg, J J Nieto & Y Zhou Vol 9: Nonlinear Waves: A Geometrical Approach by P Popivanov & A Slavova Vol 8: The Linearised Dam-Break Problem by D J Needham, S McGovern & J A Leach Vol 7: The “Golden” Non-Euclidean Geometry: Hilbert’s Fourth Problem, “Golden” Dynamical Systems, and the Fine-Structure Constant by A Stakhov & S Aranson, Assisted by S Olsen Vol 6: An Introduction to Pseudo-Differential Operators Third Edition by M W Wong Vol 5: Asymptotic Behavior of Generalized Functions by S Pilipović, B Stanković & J Vindas More information on this series can be found at http://www.worldscientific.com/series/saac Series on Analysis, Applications and Computation – Vol 12 ISAAC World Scientific 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 of Congress Cataloging-in-Publication Data Names: Cheung, Wing-Sum (Mathematician), author Title: Metric space topology : examples, exercises and solutions / Wing-Sum Cheung, the University of Hong Kong, Hong Kong Description: New Jersey : World Scientific, [2024] | Series: Series on analysis, applications and computation, 1793-4702 ; Vol 12 | Includes bibliographical references and index Identifiers: LCCN 2023023176 | ISBN 9789811266973 (hardcover) | ISBN 9789811266980 (ebook for institutions) | ISBN 9789811266997 (ebook for individuals) Subjects: LCSH: Metric spaces Problems, exercises, etc Classification: LCC QA611.28 C44 2024 | DDC 514/.325 dc23/eng20230919 LC record available at https://lccn.loc.gov/2023023176 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Copyright © 2024 by World Scientific Publishing Co Pte Ltd All rights reserved This book, or parts thereof, may not be reproduced in any form 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 of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA In this case permission to photocopy is not required from the publisher For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/13160#t=suppl Desk Editors: Nimal Koliyat/Lai Fun Kwong Typeset by Stallion Press Email: enquiries@stallionpress.com Printed in Singapore To my parents in heaven This page intentionally left blank Preface This is a volume on the Topology of Metric Spaces, which is a sub- ject the author has been teaching in the last couple of decades It can serve as a textbook or a reference on the subject concerned, mainly for advanced level undergraduates concentrating on Mathematics, Physics, Economics and Finance, etc A common serious drawback of most existing references/textbooks at this level is the lack of worked examples/exercises with detailed solutions So the readers are frequently frustrated for not being able to solve the exercises or not being able to tell whether their solutions are valid To overcome this drawback, it is the purpose of the present volume to provide plenty of worked examples and exercises, including True-or-False type questions and open-ended questions, with detailed solutions True-or-False type questions and open-ended questions are particularly effective in helping the readers to get a birds eye view of the subject and to master the materials They are also instrumental in nurturing the mathematical insight and the mathematical maturity of the readers The emphasis of most standard textbooks or reference books on the subject is the abstract and theoretical aspect, but such an approach may not be readily digestible by most readers In order to master a new concept, it is most effective to first keep in mind a few simple, concrete and familiar examples, and then when dealing with a problem in the general setting, consider first the problem in such simple concrete settings In many situations, the treatment of a problem in a simple setting could serve as a clue to tackle the problem in the general setting On the other hand, pictorization or visualization of abstract problems into simple pictures is very often instrumental to the rigorous treatment of the problems However, this is rarely provided in most standard vii viii Metric Space Topology: Examples, Exercises and Solutions references In view of this, in this volume, within each section in each chapter, blended with the concise and rigorous treatments on the materials and before tackling problems in the more abstract settings, there will be a number of concrete examples/exercises, supplemented with simple pictorizations as appropriate, to help the readers master the concepts The learning outcomes of the volume include: • Demonstrate knowledge and understanding of the basic features of mathematical analysis and point set topology (e.g., able to identify objects that are topologically equivalent) • Apply knowledge and skills acquired in mathematical analysis to analyze and handle novel situations in a critical way (e.g., able to determine whether a specific function is uniformly continuous) • Think creatively and laterally to generate innovative examples and solutions to non-standard problems (e.g., able to construct counterexamples to inaccurate mathematical statements) • Acquire sufficient background for further studies in Functional Analysis, Real Analysis, Complex Analysis, Differential and Algebraic Geometry, Probability Theory, Mathematical Physics, Engineering, Economics and Finance, etc Throughout the 30+ years of teaching experience of the author, the most frequently asked question is “What is the use of the subject?” To be absolutely frank, many branches of pure mathematics do not find any direct application in the real world whatsoever In fact, except for a small portion of the students who would go on for further studies and take mathematics research as their careers, most theorems or results they have learnt in their mathematics courses will not be seen or used for the rest of their lives So it is perfectly fine for them to forget all the mathematics contents they have learnt at school or in college But one thing they should never forget is the study process of and the way of thinking in Preface ix mathematics, that is, the mathematics training they have undergone It is not the mathematical contents they have learnt that are important but the mathematics training they received Proper mathematics training nurtures logical arguments, analytical power, critical thinking, and rational thinking With these attributes, one could excel in any career one may go into So the target of this volume is, through numerous worked examples and exercises, to guide the readers argue logically, think critically, analyze various possible scenarios, and to nurture a mathematical and logical mind that is able to tackle novel and ill-posed problems they may encounter in the future With that, the so-called life long self-learning would no longer be a burden but instead, would turn into a basic instinct The volume is perfect as a textbook or a reference for a one- semester course on Introduction to Metric Space Topology It is the intention of the author to make it concise instead of stuffing it with unessential materials, as the latter may distract the readers from the natural flow of the development of the subject Less is more For advanced undergraduate level students, mastering the central theme of a subject would be much more beneficial than dabbling in a wider range of materials shallowly and prematurely Finally, the author would like to express his gratitude to the many students who have taken his course on the subject Their enthusiastic comments and feedback on the teaching materials have provided strong incentive for his determination to write up this book The professional assistance rendered by the staff of World Scientific, especially Lai Fun Kwong, is also gratefully acknowledged