Introductory topology : exercises and solutions

261 Pages·2014·1.465 MB·English

by  Mohammed Hichem Mortad

INTRODUCTORY
TOPOLOGY
Exercises and Solutions

INTRODUCTORY
TOPOLOGY
Exercises and Solutions
Mohammed Hichem Mortad
University of Oran, 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 of Congress Cataloging-in-Publication Data
Mortad, Mohammed Hichem, 1978– author.
Introductory topology : exercises and solutions / by Mohammed Hichem Mortad (University of
Oran, Algeria).
pages cm
Includes bibliographical references and index.
ISBN 978-9814583817 (pbk : alk. paper)
1. Topology--Problems, exercises, etc. I. Title.
QA611.M677 2014
514.076--dc23
2014003079
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2014 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.
Printed in Singapore
Preface
Topology is a major area in mathematics. At an undergraduate
level, at a research level, and in many areas of mathematics (and even
outsidemathematicsinsomecases),agoodunderstandingofthebasics
of the theory of general topology is required. Many students find the
course "Topology" (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
theendweknowthatwehavetolivethereandgetusedtoit. Theyare
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 R
needs to remain true in an arbitrary topological space. For instance,
there are convergent sequences which have more than one limit, the
identitymappingisnotalwayscontinuous,anormallyconvergentseries
need not converge (although the latter is not within the scope of this
book). Soinmanyreferences, theyusetheword"usualtopologyofR",
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,
somethingthatitisnoteasytofindinanothermanuscript onthesame
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
mathematicsisbydoingexercises, theworstwayofdoingexercisesisto
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 fishing, then you have given him
food for everyday. So we hope the students are going to learn to "fish"
using this book.
v
vi PREFACE
The present manuscript is mainly intended for an undergraduate
course in general topology. It does not include algebraic and geometric
topologies. Other topics such as: nets, topologies of infinite products,
quotient topology, first countability, second countability and the T
i
separationaxiomswithi= 0,3,4,5arenotconsideredeitherorarenot
given much attention. It can be used by students as well as lecturers
andanyonewhoneedsthebasictoolsoftopology. Teaching thiscourse
several times with manydifferent exercises each year hasallowed meto
collect all the exercises given in this book. I relied on many references
(I cannot remember all of them but most of them can be found in the
bibliography) in lectures and tutorials. If there is some source which I
have forgotten to mention, then I sincerely apologize for that.
Let us now say a few words about the contents of this book. The
exercises on the subjects covered in this book can be used for a one
semester course of 14 weeks.
The bookis divided into two parts. In Part 1, each chapter (except
for the first one) contains five sections. They are:
(1) What You Need to Know: In this part, we briefly recall the
essential of notions and results which are needed for the exer-
cises. No proofs are given. We just note that this part cannot
inanycasereplaceadetailedcourseonthesubject. Thereader
may also wish to consult the following references for further
reading: [1], [2], [3], [6], [7], [9], [10], [11], [13], [14], [15] and
[16].
(2) True or False: In this part some interesting questions are
proposed to the reader. They also contain common errors
whichappearwithdifferentstudentsalmosteveryyear. Thanks
to this section, students should hopefully avoid making many
silly mistakes. This part is an important back-up for the
"What You Need to Know" section. Readers may even find
some redundancy, but this is mainly because it is meant to
test their understanding.
(3) Exercises with Solutions: The major part and the core
of each chapter where many exercises are given with detailed
solutions.
(4) Tests: This section contains short questions given with just
answers or simply hints.
(5) More Exercises:Inthispartsomeunsolvedexercisesarepro-
posed to the interested reader.
In Part 2, the reader finds answers to the questions appearing in the
section "True or False" as well as solutions to Exercises and Tests.
PREFACE vii
The prerequisites to use this book are basics of: functions of one
variable (some of several variables calculus is also welcome though not
very much), sequences and series, and set theory.
Since the terminology in topologyis rich and may be different from
a book to another, we do encourage the readers to have a look at the
"Notations and Terminology" chapter to avoid an eventual confusion
or ambiguity with symbols and notations.
Beforefinishing, I welcome and I will bepleased to receive anysug-
gestions, questions (as well as pointing out eventual errors and typos)
from readers at my email: [email protected].
Last but not least, thanks are due in particular to Dr Lim Swee
Cheng and Ms Tan Rok Ting, and all the staff of World Scientific
Publishing Company for their patience and help.
Oran on September the 24th, 2013
Mohammed Hichem Mortad
Department of Mathematics
Faculty of Exact and Applied Sciences
The University of Oran (Algeria)

Contents
Preface v
Notation and Terminology xiii
0.1. Notation xiii
0.2. Terminology xiv
Part 1. Exercises 1
Chapter 1. General Notions: Sets, Functions et al 3
1.1. What You Need to Know 3
1.2. Exercises With Solutions 4
1.3. More Exercises 6
Chapter 2. Metric Spaces 9
2.1. What You Need to Know 9
2.2. True or False: Questions 13
2.3. Exercises With Solutions 14
2.4. Tests 19
2.5. More Exercises 20
Chapter 3. Topological Spaces 23
3.1. What You Need to Know 23
3.2. True or False: Questions 29
3.3. Exercises With Solutions 32
3.4. Tests 39
3.5. More Exercises 40
Chapter 4. Continuity and Convergence 45
4.1. What You Need to Know 45
4.2. True or False: Questions 48
4.3. Exercises With Solutions 50
4.4. Tests 56
4.5. More Exercises 57
Chapter 5. Compact Spaces 61
5.1. What You Need to Know 61
ix