Mortad, mohammed hichem introductory topology exercises and solutions world scientific (2017)

Ltd.5 Toh Tuck Link, Singapore 596224USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HELibrary o f Congress Ca

I N T R O D U C T O R Y TOPOLOGYExercises and Solutions

Second Edition

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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 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.

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Printed in Singapore

Preface

Preface to the Second Edition

ixxiii

CONTENTS

vii

CONTENTS viii

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 n Topologyn (at least in the beginning) a bit confusing and not too easy (even hard for some o f 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 in­stance, 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 topol­ogy 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

I X

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 o f infinite products, quotient topology, first countability, second countability and the T* separation axioms with i = 0,3 ,4 , 5 are not considered neither or are not given much attention It can also be used by instructors and anyone who needs the basic tools of topology Teaching this course several times with many different exercises each year has allowed me to 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

[16], [18], [19], [20], [25], [26], [28] and [29],

(2) True or F a lse : In this part some interesting questions are proposed to the reader They also contain common errors which appear with different students almost every year 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) E x e rcise s w ith S o lu tio n s : The major part and the core

of each chapter where many exercises are given with detailed solutions

(4) T e s t s : This section contains short questions given with just answers or simply hints

(5) More E x e rcise s : In this part some unsolved exercises are pro­posed 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 xi

The prerequisites to use this book are basics of: functions of one variable (some of the several variables calculus is also welcome), se­quences and series

Since the terminology in topology is rich and may be different from

a book to another, we do encourage the readers to have a look at the nNotations and Terminology” chapter to avoid an eventual confusion

or ambiguity with symbols and notations

Before finishing, I welcome and I will be pleased to receive any sug­gestions, questions (as well as pointing out eventual errors and typos)

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 The University of Oran (Algeria).

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Preface to the Second Edition

In this second edition, many changes have been made, some quite significant For instance, this new edition contains proofs (as exercises)

of many results in the old section "What You Need To Know", which has been renamed here as "Essential Background" This new edition has also many additional exercises

There is another improvement still about the section "Essential Background" Indeed, it has been considerably beefed up as it now in­cludes more remarks and results for readers’ convenience As an illus­trative point, we have enlarged the subsection "Fixed Point Theorem" and even added one on applications to integral equations

I have also added other notions e.g on the topology of an arbitrary product of topological spaces as well as separation axioms Despite all these changes, the book still does not cover concepts o f Algebraic Topology

The sections "True or False" and "Tests" have remained as they were apart from a very few changes

I have also corrected typos and mistakes which appeared in the earlier edition of this manuscrit

It is worth noticing that some o f these points were suggested in the

M AA review by Dr Mark Hunacek I take this opportunity to thank him for his review

Readers wishing to contact me may do that via my email:

mhmortad@gmail.com.

Finally, warm thanks are due to all the staff of World Scientific Publishing Company for their patience and help (in particular, Dr Lim Swee Cheng and Ms Tan Rok Ting)

xm

Oran, Algeria

M a y 08, 2016

P rof D r M H Mortad

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Notation and Terminology

• Q is the set of rational numbers

• R is the set of real numbers

• C is the set of complex numbers

• [a, b] is the closed interval with endpoints a and b.

• (a, b) is the open interval with endpoints a and b.

• The power set of set X is denoted by V (X ) or 2X

• (a, b) also denotes an ordered pair

• i the complex square root of —1

• x 1-)- ex the usual exponential function

• The complement of a set A in X is often denoted by Ac and occasionally it is denoted by X \ A or X — {a }

• The empty set is denoted by 0

• d(x, A) is the distance between a point x in X to a set A C X

where (X , d) is a metric space

• d(A) is the diameter of a set A in a metric space (X, d).

• In a metric space (X, d), the open ball of radius r > 0 and center x E X is denoted by B (x ,r).

• In a metric space (X ,d), the closed ball of radius r > 0 and center x E X is denoted by Bc(x,r).

• In a metric space (X, d), the sphere of radius r > 0 and center

x G X is denoted by S (x,r).

o

• The interior of a set A in a topological space is denoted by A.

• The closure o f a set A in a topological space is denoted by A.

• A f denotes the derived set of A, that is the set of limit points

of A (where A is a subset of a topological space X ).

• The frontier of a set is denoted by Fr or d.

• card denotes the cardinal of a set

X V

NOTATION AND TERMINOLOGY

• R^ is the lower limit topology

• The real part of a complex number is denoted by Re

• The imaginary part of a complex number is denoted by Im

• The conjugate of a complex number z is denoted by z.

• C ([0,1]) (or C ([0,1],R )) is the space of real-valued continuous functions on [0 ,1] taking values in R If the field o f values is

C, then this will be clearly mentioned

• C,1([0 ,1]) (or C1([0 ,1], R )) is the space of real-valued contin­uous functions defined on [0,1], differentiable and having a continuous derivative

• (R, | • |) is R equipped with the standard or the usual metric,i.e the absolute value function

• It will be comprehensible from the context whether "c" is for comparing two sets or two topologies

• It will be clear from the context whether (a, b) is the ordered pair or the open interval

• When it is not too important to specify the metric or the topology, we simply say the topological (or metric) space X

• From time to time the reader will see "(w hy?)" In such case, this means that this is a question whose answer should be known by the reader This is used by other authors such as

J B Conway (see [7]) and also M Stoll (see [28]) Other expressions such as "(is it not?)" are also used

• The letters "w.r.t." stand for "with respect to"

• As it is used almost everywhere, "iff" means "if and only if" For the fun, the French use "ssi" for "si et seulement si" Even

in Arabic, it has been sorted out by doubling a letter in the end!

0.2 TERMINOLOGY

As "iff” , there are other words which cannot be found in an English language dictionary but we still use them For exam­ple, "Cauchyness", "Hausdorffness", "clopen", etc

W LOG, as it pleases many, stands for "without loss of gener­ality"

If readers see in the end of an answer "•••", then this means that details are left to them

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Part 1

Exercises

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1.1.1 Sets We start with the definition of a set We just adopt the naive definition of it as many textbooks do We shall not go into much detail of Set Theory It can be a quite complicated theory and

a word from J M Mpller suffices as a warning He said in [18]: "We don’t really know what a set is but neither do the biologists know what life is and that doesn’t stop them from investigating it" So,

we do advise the student to be careful when dealing with set theory

in order to avoid contradictions and paradoxes Before commencing,

we give just one illustrative paradox which is the R u ss e l’s set (from

R u sse l’ s pa ra d ox : if someone says "I’m lying", is he lying?): Let

Z, etc Elements or objects belonging to the set are usually designated

by lowercase letters: a, b, c; x, y, z, etc

that a 0 A.

If a set does not contain any element, then it is called the em pty

to denote it by U).

Ex a m p l e s 1.1.2

(1) { —4 ,1,2} is a set constituted of the elements 1, 2 and —4

elements.

4 1 GENERAL NOTIONS: SETS, FUNCTIONS ET AL.

One element sets (e.g {2}) have a particular name

DEFINITION 1.1.3 A singleton is a set with exactly one element.

The notion of a "smaller” or "larger" set cannot be defined rigourously even though we will be saying it from time to time We have a substi­tute

De f in it io n 1.1.4 Let A and B be two sets.

A ) if every element of A is an element of B, and we write

inclusion.

A c B and B C A.

inclusion.

Ex a m p l e 1.1.5 We know that:

N c Z c Q c R

There are other special and important subsets of the real line R

DEFINITION 1.1.6 A subset I of R is called an interval if:

half-open intervals (or half-closed, why not?!).

[a, 00) = {x G R : x > a} or (—00, b) = {x G R : x < b}.

EXAMPLES 1.1.7 (details are to be found in Exercise 1.2.8)

(1) Q or R \ Q are not intervals.

(2) M* is not an interval.

(3) [0,1] U { 2 } is not an interval.

We can produce new sets from old ones using known operations in Mathematics

De f in it io n 1.1.8 Let A and B be two sets.

A n B = {x : x E A or x e B }.

by

A D B = {x : x E A and x E B }.

A — B = {x : x e A and x ¢ B }.

We list some straightforward properties

Pr o p o s it i o n 1.1.9 Let A and B be two sets Then:

DEFINITION 1.1.10 Let A be a subset of a set X Then the set

X — A is called the com plem ent of A, and it is denoted by Ac.

REMARK It is useful to keep in mind that:

x E Ac <£=> x 0 A.

6 1 GENERAL NOTIONS: SETS, FUNCTIONS ET AL.

REMARK We can re-write A \ B using complement We have indeed:

The next easy two results are fundamental

Pr o p o s it i o n 1.1.12 Let A and B be two sets Then

A c B <(=> B c c Ac.

Pr o p o s it i o n 1.1.13 Let A and B be two sets Then

A n B = 0 <<=> A c B c <<=> B c Ac.

Going back to intervals, natural questions are: Is the intersection

of two intervals an interval? What about their union?

Th e o r e m 1.1.14

REMARK In the second statement of the previous theorem, the hypothesis on the intervals not being disjoint is indispensable as is seen

by considering the intervals [0,1] and [2,3]

Before passing to rules of set theory, we have yet two more ways of producing new sets from old ones Here is the first one:

De f in it io n 1.1.15 Let A and B be two sets The cartesian product of A and B, denoted by A x B, is defined to be:

A x B = {(a , b) : aC A, b C B },

If A = B, then we write A x A = A2.

REMARK It will be clear from the context whether (a, b) denotes

an ordered pair or an open interval For this reason, I prefer the French notation (for instance) for an open interval which is ]a, b[.

REMARK We can extend the concept of a cartesian product to a finite number of sets Ai, i = 1, • • • , n. It is given by

A\ x • • • x An = {(a i, • • • , an) : a* E Ai for all i}.

This may also be denoted by YYi=i

1.1 ESSENTIAL BACKGROUND 7

Ex a m p l e 1.1.16 Let A = { 0 ,1 ,2 } and B = { a , /3} Then

A x B = { (0, a ), (1, a), (2, a ), (0, /3), (1, /3), (2, /3)}

DEFINITION 1.1.17 Let X be a set The set of all possible subsets

denote it by V( X)

REMARK In some references, they denote the power set by 2X

(the reason of this notation will be known in Theorem 1.1.19)

Ex a m p l e 1.1.18 Let X = {a,b} Then

V ( X ) = { 0 , { a } , { b } , X }

Re m a r k s

(1) The readers should not forget that the elements of V ( X ) are sets! So if X = {a, &}, then e.g we write { a } E V ( X ) and not

we are allowed to write { { a } } C V( X)

(2) We have seen that if X has two elements, then V ( X ) has four Can we generalize this? We can’t yet! Indeed, a set of 2

elements gives a power set o f 4 elements So if a set is of p

elements, then is its power set of 2P or p2 elements? Let us therefore pass to three elements Let Y = {a, 6, c} Then

V( Y) = {0, { a } , {6}, { c } , {a, &}, {a, c}, {&, c}, X }

so that V( Y) has now 8 elements, i.e it has 23 elements In fact, this is true for any n as seen next

THEOREM 1.1.19 (see Exercise 1.2.10) Let X be a set having ”n "

elements.

In many situations in mathematics we will have to take unions mixed with intersections (and sometimes products as well) How to remedy this?

Th e o r e m 1.1.20 Let A , B , C and D be sets Then:

1 GENERAL NOTIONS: SETS, FUNCTIONS ET AL.

DEFINITION 1.1.21 Let (A i ) ieI be a family of subsets of X , where

As in the case of finite unions and intersections, distributive and

De Morgan’s laws remain valid in the setting of an arbitrary class

of sets More details are given next

Ai = {x e X : x E Ai at least for one i E / }

=UxeM]{a;}-THEOREM 1.1.23 Let ( be a family of subsets of X , where I

( i )

(2)

( 3 )

(4)

1.1 ESSENTIAL BACKGROUND 9

Lastly, we give a result which is easy to show (yet a proof is given

in Exercise 1.2.9) We will be calling on it on many occasions in the present book

PROPOSITION 1.1.24 Let A be a set Assume that for each a G A,

A = \ J U a.

aeA

1.1.2 Functions We give the most basic definition of a function without going into deep details

De f in it io n 1.1.25 Let X and Y be two sets A function f (or

mapping or even map) from X to Y is a rule which assigns to each

De f in it io n 1.1.26 Two functions f : X ->► Y and g : X Z

f ( x ) = g ( x ) , V x e X

DEFINITION 1.1.27 The identity function on a s e t X is the func­ tion f : X —> X defined by f ( x ) = x It is usually denoted by id x•

A related (and different) example is:

De f in it io n 1.1.28 Let A c X The inclusion map on a A is

the function l : A X defined by l(x )= x.

An inclusion map is a restriction of the identity in the sense given next

De f in it io n 1.1.29 Let f : X - > Y be a function and let A c X

f ( x ) = g(x), Vx G A.

We now give two important subsets of the domain and the codomain

of a given function

De f in it io n 1.1.30 Let f : X - * Y be a function Let A c X and B c Y Then:

10 1 GENERAL NOTIONS: SETS, FUNCTIONS ET AL.

(1) The image (or range) of A, denoted by f ( A) , is defined to be:

(4) The reader should bear in mind that in general f ( X ) C Y.

(5) It is evident that for any / : / -1(T ) = X.

(6) The inverse image is defined for any function / , i.e / is not assumed to be invertible (see Definition 1.1.37 below), even though in case of invertibility we still denote the inverse by

/ -1 (in such case the two notations coincide)

EXAMPLES 1.1.31 Let f : M —> R be defined by f ( x ) = x 2 Then:

THEOREM 1.1.33 (see Exercise 1.2.11) Let I be an arbitrary set

-DEFINITION 1.1.34 Le£ X , y and Z be three non-empty sets Let

com posite) of g with f is the function denoted by g o f f and defined from X into Z by

( j ° / ) ( * ) = £ ( / ( * ) ) •

De f in it io n 1.1.35 Let f : X X be a function The iterated

by composing f with itself "n " times In other words, we define f ^ as follows:

/ (0) = idx , f (n+1) = f (n) °

/-Pr o p o s it i o n 1.1.36 (see Exercise 1.2.11) Let f : X -¥ Y and

g : Y -¥ Z be two functions and let A c Z Then

( g o f ) - \ A ) = f - l (g -\ A )).

REMARK It is worth noticing that no n is not commutative but it

is associative It is also easy to see that for all /

f o i d = i d o f = f

where id is the identity function on X. The next natural question is: When is a function invertible? We state this definition separately

De f in it io n 1.1.37 A function f : X - > Y is invertible, if there

f o g = iY and g o f = ix , where idx and idy are the identity functions on X and Y respectively

In other words, f is invertible if

( / ° 9){y) = V cmd ( / o g)(x) = x

12 1 GENERAL NOTIONS: SETS, FUNCTIONS ET AL.

REMARK In case of invertibility, we may write (x 6 X and y E Y)

f ( x ) = y <=> x = f - 'i y )

Next, we introduce concepts which help us to check invertibility of functions

De f in it io n 1.1.38 Let f : X - > Y be a function We say that f

is injective (or one-to-one) if:

We have just noticed that f ( X ) does not always equal Y. But when

it does, this has the following appellation

De f in it io n 1.1.39 Let f : X Y be a function We say that f

is surjective (or onto) if:

Vy G Y, 3x e X : f ( x ) = y,

or equivalently if f ( X ) = Y.

REMARK The word "onto" may be a little bit misleading for peo­ple who have not done their first degree in English They might think that a function from X onto Y is just like a function from X to Y\

This is of course not true as we have just said that: onto=surjective!The "same expression" of f ( x ) may produce each time a different result as regards surjectivity and injectivity if we change the domain and the codomain of / :

Ex a m p l e s 1.1.40 Let f ( x ) = x2 Then

(1) / : R —> E is neither injective nor surjective;

(2) / : R + —► R is injective but not surjective;

(3) / : K —)• M+ is surjective but not injective;

(4) / : M+ —► R + is injective and surjective.

It can happen that a function is injective and surjective at the same time (as in the last example) This leads to the following fundamental notion

1.1 ESSENTIAL BACKGROUND 13

DEFINITION 1.1.41 Let f : X —» Y be a function We say that

surjective simultaneously, that is, iff

My c Y , 3 \ x c X : f ( x ) = y

One primary use of bijective functions is invertibility

THEOREM 1.1.42 A function is invertible if and only if it is bijec­ tive.

We now give properties of composite functions as regards injectivity and surjectivity

THEOREM 1.1.43 (see Exercise 1.2.16) Let X , Y and Z be three

(1) If f and g are injective, then so is g o /

THEOREM 1.1.44 (see Exercise 1.2.15) Let X , Y and Z be three

(1) If g o f is injective, then f is injective.

COROLLARY 1.1.45 Let X , Y and Z be three non empty sets Let

f : X - > Y and g : Y —» Z be two functions Then:

g o f bijective = > f injective and g surjective.

The function defined next plays a paramount role in many areas of mathematics

De f in it io n 1.1.46 Let X be a set and let A c X The indicator

We finish with another useful class of functions.

DEFINITION 1.1.48 Let I be an interval We say that the function

f : I —> R is convex si

Vxi,x2 6 /, VA e [0,1] : / ( Axi + ( 1 - A)x2) < A/(xi) + ( 1 - A)/(x2).

The convexity is easily characterized using differentiability

THEOREM 1.1.49 If f : I —> R is a twice differentiable function, then

f is convex -<=>- f " > 0

EXAMPLE 1.1.50 The real-valued function given by f ( x ) = — lux

Another (important) example is stated separately:

PROPOSITION 1.1.51 Let a e l Then x eax is convex on R.

Finally, we recall two of the fundamental theorems in basic real analysis

Th e o r e m 1.1.52 (Intermediate Value Theorem) Let f : [a, b]

—)-R be a continuous function Assume that f(a) < f(b) Then

V7 6 [ / ( a ) , / ( 6 ) ] , 3c E (a, b) : f { c ) = 1

REMARK For a generalization of the previous result see Corollary 6.1.9

Th e o r e m 1.1.53 (Mean Value Theorem) Let f : [a, b] —> R be

3c E (a, b) : f(b) — f (a) = (b — a) f (c )

14 1 GENERAL NOTIONS: SETS, FUNCTIONS ET AL.

1.1 ESSENTIAL BACKGROUND 15

1.1.3 Equivalence Relation.

De f in it io n 1.1.54 A relation 1Z on a set X is a subset An of

xTZy.

DEFINITION 1.1.55 Let IZ be a relation on X We say that IZ is:

(1) reflexive if: Vx G X : xlZx.

(2) sym m etric if: \/x,yG X : x1Zy ylZx.

(3) transitive if: Mx,y,z G X : xlZy and y1Zz => xTZz.

If 1Z is reflexive, symmetric and transitive, then we say that 1Z is an

equivalence relation on X

EXAMPLE 1.1.56 The relation 1Z defined by:

xlZy cos2 y + sin2 x = 1

DEFINITION 1.1.57 Let 1Z be an equivalence relation on X The

equivalent class of x G X , denoted by x (or [x]), is defined by:

The collection of all equivalent classes of X , denoted by X/1Z is

X/TZ = { x : x e X }

Now, we give basic properties of the quotient

PROPOSITION 1.1.58 Let 1Z be an equivalence relation on X and

(1) Vx G X : x G x

(2) x = y iff xlZy.

1.1.4 Consequences of The Least Upper Bound Property.

We now turn to important consequences of the Least Upper Bound Property (basic results on the least or greatest bound property are assumed)

Th e o r e m 1.1.59 (Archimedian Property) Let x ,y G R with

x > 0 Then there exists nG N such that

nx > y.

REMARK Another formulation o f the previous theorem is to say that N is not bounded from above

16 1 GENERAL NOTIONS: SETS, FUNCTIONS ET AL.

As a first application of the previous theorem, we introduce a useful class of function

De f in it io n 1.1.60 The greatest in teger fu n ction , denoted by

the largest integer "n" less or equal to x.

THEOREM 1.1.63 (see Exercise 1.2.32) The set of rational num­

a rational one.

Similarly, we can prove the following:

THEOREM 1.1.64 (see Exercise 1.2.32) The set of irrational num­ bers R \ Q is dense in R, that is, between any two real numbers, there

is an irrational one.

Re m a r k s

(1) The precise meaning of density will be made clear in its general context in Chapter 3

(2) The two previous results are usually expressible respectively

as follows (the reason will also be known in Chapter 3):(a) Any real number is the limit of a sequence of rational numbers

(b) Any real number is the limit of a sequence of irrational numbers

Another important consequence of the Least Upper Bound Property

is the following (a proof is prescribed in Exercise 1.2.25):

Th e o r e m 1.1.65 (N ested Interval P roperty) Let (7n)n be a

h = [^2> b<f\, such that / i D /2 Z> • • • Then

nGN

1.1 ESSENTIAL BACKGROUND 17

that is, there is at least one point common to each interval In.

REMARK The fact that the intervals are closed and bounded is crucial for this result to hold See Exercise 1.2.27

This result will be generalized in Corollary 5.1.6 (cf Theorem 7.1.27)

We finish with a consequence of Theorem 1.1.65 about sequences

Co r o l l a r y 1.1.66 (Bolzano-Weierstrass Property , cf Def­

quence.

1.1.5 Countability.

DEFINITION 1.1.67 Two sets A and B are said to be equivalent,

and we write A ~ B, if there a bijection between A and B.

REMARK The concept of equivalence of sets is an equivalence relation

DEFINITION 1.1.68 We say that a set A is finite if it is empty or

if there is a bijection f : A {1 ,2 , • • • , n} (i.e A ~ { i , 2 ,• • • ,«> ; if

Ex a m p l e s 1.1.69

(2) If A = 0 , then card^4 = 0

(3) If A = { { 0 } } , then card^4 = 1

THEOREM 1.1.70 If a set A is finite, then there is no bijection of

A with any proper subset of A.

Ex a m p l e 1.1.71 The set N is infinite.

De f in it io n 1.1.72 We say that a set A is countable (or denu­ merable) if it is finite or if there is a bijection f : A —> N A set which

18 1 GENERAL NOTIONS: SETS, FUNCTIONS ET AL.

(2) Z is countable.

When looking for a bijection between N and a given set A, it may happen that we find a function / which is injective but not surjective and vice versa The next result is therefore quite practical

THEOREM 1.1.74 Let A be a non-empty set Then the following statements are equivalent

PROPOSITION 1.1.75 (see Exercise 1.2.19) If X is countable, then

so is any A d X

EXAMPLE 1.1.76 The set of prime numbers is countable

Th e o r e m 1.1.77

REMARK It is plain that any intersection of countable sets is at most countable!

COROLLARY 1.1.78 The set of rationale Q is countable.

It is natural to ask whether M is countable?

THEOREM 1.1.79 (see Exercise 1.2.22 or Exercise 1.2.29) The

Co r o l l a r y 1.1.80 R is uncountable.

REMARK Another yet simpler proof may be found in Exercise7.3.35

Co r o l l a r y 1.1.81 R \ Q is uncountable.

1.2 EXERCISES WITH SOLUTIONS 19

The previous result may be generalized to:

PROPOSITION 1.1.82 Let X be an uncountable set and let A C X

be countable Then X \ A is uncountable.

THEOREM 1.1.83 ([19]) Let A be a non-empty set, and letV(A) be the set of all subsets of A Then there is no injective map f : V(A) —>

A Also, there is no surjective map g : A -¥ V(A).

Co r o l l a r y 1.1.84 V(N) is uncountable.

We finish this section with an important set in analysis To define

it, consider first the set Ao = [0,1] The set A\ is obtained from A$ by removing the middle third interval (|, |) To obtain A2, remove from

A\ its middle thirds, namely (^, §) and (|, |) Then set

In Measure Theory, the Cantor set constitutes an example of an uncountable set with zero Lebesgue measure

1.2 E x ercises W it h S olu tion s

E x e rcise 1.2.1 Show that \/2 is irrational

E x e rcise 1.2.2 Show that

In 2

E x e rcise 1.2.3 Show that e 0 Q

Hint: you may use the sequences defined by

20 1 GENERAL NOTIONS: SETS, FUNCTIONS ET AL.

B-(3) A n B = A.

Exercise 1.2.5 Let A , B , C be three sets Assume that A C B

Does it follow that

Exercise 1.2.7 Let A and B be two subsets of X. Show that A f) B

Exercise 1.2.8 Show that none of the following sets is an interval:

Q, R \ Q, M* and [0,1] U {2 }

Exercise 1.2.9 Let A be a set Assume that for each a G A, there

is a subset Ua of A containing a, i.e a E Ua C A. Show that

Exercise 1.2.11 Let I be an arbitrary set Let / : X —» Y and

g : Y —> Z be two functions Also assume that A, Ai C X; B, Bi C Y

and C C Z. Show that the following statements hold

-1.2 EXERCISES WITH SOLUTIONS 21

(8) C / _1( /( A ) ) Do we have A D f ~ 1(f(A))7 (cf Exercise1.3.3)

Exercise 1.2.12 Let f : X Y be a function If is the re­striction of / to C X , then show that for any subset U of X one has

(1) Show that if g o f is injective, then / is injective

(2) Show that if g o f is surjective, then g is surjective

(3) Infer that

g o f bijective = > / injective and g surjective

Exercise 1.2.16 Let X , Y and Z be three non-empty sets Let / : X Y and g : Y —> Z be two functions Show that:

(1) If / and g are injective, then so is g o /

(2) If / and g are surjective, then so is g o /

Exercise 1.2.17 Define a relation 1Z on R by

for all x, y E R Show that 7Z is an equivalence relation on R