0198702337 Topology c

Undergraduate Topology

UndergraduateTopologyAWorking Textbook

AISLING McCLUSKEYSenior Lecturer in MathematicsNational University of Ireland, Galway

BRIAN McMASTERHonorary Senior LecturerQueens University Belfast

3

3Great Clarendon Street, Oxford, OX2 6DP,United KingdomOxford University Press is a department of the University of Oxford.It furthers the Universitys objective of excellence in research, scholarship,and education by publishing worldwide. Oxford is a registered trade mark ofOxford University Press in the UK and in certain other countries Aisling McCluskey & Brian McMaster 2014The moral rights of the authors have been assertedFirst Edition published in 2014Impression: 1All rights reserved. No part of this publication may be reproduced, stored ina retrieval system, or transmitted, in any form or by any means, without theprior permission in writing of Oxford University Press, or as expressly permittedby law, by licence or under terms agreed with the appropriate reprographicsrights organization. Enquiries concerning reproduction outside the scope of theabove should be sent to the Rights Department, Oxford University Press, at theaddress aboveYou must not circulate this work in any other formand you must impose this same condition on any acquirerPublished in the United States of America by Oxford University Press198 Madison Avenue, New York, NY 10016, United States of AmericaBritish Library Cataloguing in Publication DataData availableLibrary of Congress Control Number: 2014930102ISBN 9780198702337 (hbk.)ISBN 9780198702344 (pbk.)Printed and bound byCPI Group (UK) Ltd, Croydon, CR0 4YY

Preface

This textbook oers an introduction at undergraduate level to an area knownvariously as general topology, point-set topology or analytic topology, with a par-ticular focus on helping students to build theory for themselves. It emerged as aresult of several years of our combined university teaching experience, stimulatedby sustained interest in advanced mathematical thinking and learning, alongsideestablished research careers in analytic topology. The modern-day student andinstructor can benet hugely from a contemporary text in the subject, a text thatseeks to get to the heart of individual and independent mathematical learning.Our long experience suggests that there is a need for such a text.

Point-set topology is a discipline which needs relatively little background know-ledge, but a good measure of determination to grasp the denitions precisely andto think and argue with straight and careful logic. It is widely acknowledged thatthe ideal way to learn such a subject is to teach it to yourself, proactively, by guidedreading of brief skeleton notes and by doing your own spadework to ll in the de-tails and to esh out the examples. In reality, many students simply do not have thecombination of time and commitment to implement this plan fully. A gentle touchis increasingly needed to engage students with the material and to build their con-dence in handling it. Students undertaking this subject must come to terms withhigher levels of abstraction and more sophisticated proofs than they are likely tohave encountered before, and for such challenging notions it is invaluable to havean expert guide on hand. This text oers the possibility of step-by-step learningwith relevant help, where required, in close proximity within the text. Thus it ad-dresses a broad range of needs as reected in the typically diverse student body: itfacilitates the captive student who is obliged to take the subject as a prerequisitefor some other purpose, the student who does have time and commitment (evenfor a modest segment of the teaching term), and the instructor who would like toencourage the so-called Moore approach (again, even for part of the time or partof the class), and it can equally well be used for more conventional instructor-ledcourses.

The writing style is characterised by the use of accessible and engaging lan-guage, critically reminiscent in sucient measure (not too much, not too little)of a workbook as opposed to the dense text that is the norm in other titles. Theintention is that it will invite the reader/learner to use it and that, in particular,it will serve as a strong basis for self-directed study. Choice of content is almostinevitable for such a book; rather, it is how the content is presented to, and poten-tially experienced by, the learner that sets this text apart from the rightly reveredclassics of the area. Here we oer an alternative style of presentation, informed bylong experience in the business of teaching and learning mathematics and also byresearch within the eld of mathematics education, particularly in the area of ad-vanced mathematical thinking. Even the visual appearance of a formal argument

vi PREFACE

upon the printed page can now be a signicant barrier to engagement for many.We have sought to overcome this by favouring an approach that mirrors the slow,careful process of developing a mathematical argument rather than opting for amore stylish monograph presentation. More specically, it is our aim to facil-itate a route to learning that enables comprehension, and therefore condence,competence and enjoyment of the subject, and that wrests memorisation from itspotentially dominant and devastating position in contemporary undergraduatemathematics education. The text oers content that is limited to the fundamen-tal concepts of the subject, presented in such a fashion that students may achievedepth in their understanding of and facility with that content. The drive is forquality rather than quantity.

In essence, this is a text written by twomathematicians who are passionate bothabout the subject area and about teaching students. It is a result of long familiaritywith the mathematical learning process, particularly in this subject area and withthe contemporary student.

Contents

Comments to the instructor: what types of study this book will support ixComments to the student reader: how to use this book xi

1 Introduction 1

Preamble 1

Sets and mappings 3

Zorns lemma 5

The least uncountable ordinal 8

Metric spaces 8

Expansion of Chapter 1 13

2 Topological spaces 17

Some elementary concepts 17

Subspaces 20


3 Continuity and convergence 29

Continuity 29

Convergent sequences 32

Nets 35

Filters 36


4 Invariants 49

Sequential compactness 49

Compactness 50

Local compactness 52

Connectedness 52

Separability 54


5 Base and product 64

Base 64

Complete separability 65

Product spaces 66

Product theorems 68

Infinite products 69


6 Separation axioms 85

T1 spaces 85

T2 spaces 86

viii CONTENTS

T3 spaces 88

T3 12

spaces 88

T4 spaces 89


Essential Exercises 106Solutions to selected exercises 118Suggestions for further reading 141Index 143

Comments to the instructor: what typesof study this book will support

There are many excellent textbooks on general topology in print; what need isthere of another?

The idea for this text arose from a related question that we occasionally askedourselves: there are many excellent textbooks on general topology in print; whydo we not teach out of one of them, rather than painstakingly creating our ownlecture notes time after time?

A major part of the answer for the present authors, anyway lies in the ex-pectations and preparedness of contemporary mathematics undergraduates, who,although just as skilled and insightful as ever in maths as such, tend to be shortof experience in reading and writing extended prose passages of cumulative ar-gument and perhaps through a bias in the assessment system at post-primarylevel over-reliant upon memory and memory-driven algorithms rather thanupon fresh analysis of a question and the subsequent construction and commu-nication of a persuasive answer to it. This is a double pity: for the mathematicsgraduate is virtually certain to be faced with problems for which no pre-learnedalgorithmwill suce, andmuch of the enjoyment and excitement of our disciplinecome from the struggle with such problems.

The ideal solution may very well be implementation of some version of theMoore method, in which students are presented only with the denitions and thestatements of key results and examples, and encouraged to discover and developtheir own arguments for what is asserted. Indeed, for a special class of under-graduate, this is a perfect mechanism for enhancing key skills of comprehension,construction of logical evidence and coherent communication. Yet, realistically,many students do not have the time, the motivation and the freedom from othercompeting demands on their energies to follow this path entirely. A text that en-courages each of them to try this method for a proportion of their study, but thatstill oers a complete account of the material for when their time and energy runout, can bring benets proportionate to the investment of eort that each nds itpracticable to devote.

Our other key consideration was the choice of format in which to set forth thatcomplete account, and here we confess to a degree of sympathy with studentscommon complaint that mathematics books are dicult to read. Publishedmathematics for a professional readership is usually couched in lengthy, complexparagraphs of good literary style and rightly so, given the appropriate sophistica-tion of the argument and the target market and similar eloquence is a frequentand enjoyable feature of the well-written textbook. Yet it seems to us importantnot to create the impression that learners are expected to do likewise: on the con-trary, the aim that most of us hold for most of our students, as regards how they

x COMMENTS TO THE INSTRUCTOR

communicate their ndings, is that they will develop a facility in proceeding step-wise through initially uncharted territory, verifying each point before going on tothe next: bullet-point thinking, if not actually bullet-point writing. This exposi-tory style at least as old as Euclid has served us well for seventy generations,and it is also close to the manner in which modest mathematical discoveries areoften made, especially in axiomatic work. With this in view, we have generallyset out our demonstrations one step at a time, rather than in free-owing prose.Our experience is that this makes it easier for the learner to compare his or herown attempts with model solutions and to comprehend such models where timepressures have prevented the preparation of personal attempts, that it sets a morerealistic standard for the type of presentation that a diligent student should beproducing by about the second or third draft of an assignment, and that it encour-ages systematic thinking both in discovery and in exposition. In this way, we seekto reduce the apparent gap between a learners rst rough-work exploration of anexercise and a submittable version, incorporating into the text aspects of teach-ing more usually to be found in tutorial sessions. The downside that it uses upa little more paper than a compacted, uid discourse would have done may beconsidered a reasonable price to pay for the potential benets.

So we have prepared a text that oers, in each chapter, an expository section thatyou could opt to make the rawmaterial for aMoore treatment, and an Expansionsection giving complete specimen proofs mostly in a logical but semi-informalbullet-point style such as a good student should aspire to create that you couldequally well interleave with the exposition to develop a conventional, tutor-ledcourse.Where you position your delivery across that spectrum is for you to decide,but the book is designed to support your choice wherever it falls at each stageof the course. There is also an extensive Essential Exercises section, with modelsolutions to around half of the questions, so that you are resourced with a rangeof genuinely unseen assignments should that be appropriate. The introductoryChapter 1 summarises the background usually expected in such a class: essentially,an elementary treatment of metric spaces and the basics of analysis and of settheory that lead into such a study.

Comments to the student reader:how to use this book

Teaching topology over many years to a wide variety of students has convincedus that the perfect way to learn this material is not to be taught it at all. Learningit for yourself, through a guided reading of the key ideas and results, followed upby creating your own arguments to support what is claimed to be true either asan individual or in small study groups will give you a mastery of the subject, adepth of understanding, an ownership of the information that is very dicult toacquire quickly in any other way.

In the real world, however, most students simply do not have the combinationof time and patience that full-blown self-tuition requires. What we have set out inthe present book is a text that will allow you, the reader, to do as much self-tuitionas will t in with all the other demands on your attention, but provide a completeaccount that you, your class and your instructor can treat as a more conventionalteaching textbook as and when necessary. Nevertheless, we strongly advise you totake the self-instruction route for at least part of the time, for the reasons outlinedabove.

Each chapter consists of two sections which can usefully be thought of asExposition and Expansion. The rst part provides a collection of denitions, il-lustrations and results drawn from the standard content of a General Topologycourse, typically at nal or pre-nal undergraduate level. The second (and nor-mally larger) part sets out supporting evidence for all of the assertions made in therst, excepting those that are genuinely immediate from the denitions. Generallyspeaking, this evidence is presented not in a nalised monograph format, but in apoint-by-point fashion, rather as we would hope a diligent and time-rich studentwould set it out at about third draft. Here is how we believe you should use eachchapter for best eect:

(a) Read carefully the rst section. In particular, get the denitions very clear inyour mind, using the examples to sharpen your comprehension of them.

(b) Make time to construct proofs of as many of the assertions especially theeasier ones as you can. The material in this book is actually veryself-contained, and all the ideas necessary for each demonstration should bein what you have just read (including the earlier chapters, of course), as longas you have acquired the necessary modest background (see below) and areprepared to argue logically.

(c) Check your demonstrations against those provided in the second (theExpansion) section of the chapter, and study the other proofs it provides forthe results that you did not work out for yourself.

xii COMMENTS TO THE STUDENT READER

(d) Do several of the Essential Exercises. Again, check your solutions against thespecimen ones where these have been covered in the Solutions to selectedexercises. Read as time permits some of the other Essential Exercises andmodel solutions.

(e) Proceed to the next chapter and repeat.

In connection with background, what a student wishing to study topology atthis level needs to have got to grips with already is

(i) a decent understanding of basic real analysis;(ii) a familiarity with the language of really elementary set theory (preferably,

but not necessarily, including an encounter with the axiom of choice andZorns lemma);

(iii) an introductory course in metric spaces; and(iv) a clear understanding of what counts as a proof in mathematics.

We do, in fact, revise these matters in Chapter 1, but not at sucient length togive the reader the condence in their use that is needed in order to make goodprogress.A word about diagrams: most mathematicians working in this area, at what-

ever level of experience, draw rough-work diagrams all the time as props to theirintuition, as communication icons between themselves, as explanatory devicesfor their students and for half a dozen other reasons. We are keen to encouragestudents to doodle in this fashion at every stage of working through the book, pro-vided only that it is understood that a diagram is not a proof. Please keep in mindthat anything drawn upon a piece of paper, a blackboard or a whiteboard is ne-cessarily living inside a two-dimensional Euclidean metric space, and that most ofthe interesting questions in topology reside in spaces whose structures are muchmore complicated than that. All the same, simple and properly interpreted dia-grams are immensely valuable in grasping ideas and their interconnections, andwill often serve as steps towards discovering a proof for oneself as we hope toillustrate in a score of locations.

1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Preamble

When you rst began to study analysis whether real or complex you foundthat its central concerns were two distinct but closely interlinked ideas: limits andcontinuity.

Limits were concerned with approximation; they said that, by keeping on going,we could get better and better approximations to some ideal or ultimate position indeed, approximations that came within any required tolerance from the ideal.There were, however, many apparently dierent types of limit: limits of sequences,sums to innity of series, limits of functions f (x) as x C or as x C a oras x C a+ . . . , and later perhaps you met limits of sequences of functions, andwrestled with the subtle distinction between pointwise convergence and uniformconvergence in this case. All of these ideas were given denitions that were rec-ognisably similar in broad outline (along the lines of for all positive % there existspositive $ such that, as soon as you cross the $-threshold, all the approximationsare good enough to pass the %-test) but they diered in detail, depending on whichvariety of limit was being used. Most reasonable people are initially confused andirritated by these dierences. (Is $ an integer or must it be allowed to be a realnumber this time? Is it |x a| < $, or a < x < a + $, or x > $? Does itreally matter whether one writes L % > 0 M $ > 0 . . . or M $ > 0 such thatL % > 0 . . . ?) Yet with practice, experience and the passing of time, most of thesediculties fade. Nevertheless, the striking resemblances between the results forthese allegedly dierent limit concepts

(for instance,lim(an + bn) = lim an + lim bn,

(an + bn) =

an +

bn,limxC(f (x) + g(x)) = limxCf (x) + limxCg(x),limxCa (f (x) + g(x)) = limxCa f (x) + limxCag(x))

leave behind a feeling that they are not really dierent at all, but rather that theyare special cases of a more general denition which embraces all of them; and thatif only we had the vision (and determination) to grasp this wider limit idea, all theparticular results would fall into place as corollaries of theorems which we wouldthen be able to prove about the general one.

2 1 INTRODUCTION

Continuity, initially, was a more geometrical idea: it said that you could drawthe graph of a function without lifting your pen o the page. Admittedly, thatdenition began to overstretch when you tried to apply it to functions such as

f (x) = x sin(x1) if x 0,f (0) = 0,

but it was a reasonably useful intuitive prop for functions having R or an inter-val in R as domain. Once you wanted or needed to examine functions that werecontinuous at some points but not at others, such as [x], the integer part of x, orfunctions whose domain was not an interval, then a more precise and formal def-inition was required: and it was provided by the limit concept a map f was tobe considered as continuous on a set A if, for each point a0 of A, the limit of f (x)as x C a0 was f (a0), understanding that x has to belong to A here in order thatf (x) shall make sense. This allowed you to deal properly with functions such as theclassic Dirichlet function

f (x) =

{1 if x is rational,0 if x is irrational,

which could then be seen as continuous on the rationals, and also continuous onthe irrationals, and yet massively discontinuous on the reals. It also, of course,allowed you to prove results about continuity almost immediately using results onlimits.

The above two paragraphs have been written on the assumption that the num-bers involved were real; but virtually all the general remarks apply equally wellto the complex case. (The only signicant dierences arise from the fact thatthe complex eld is not ordered, and so one must avoid writing things likelimzCz+0 f (z) or z > z0 once z and z0 are allowed to go complex.) Indeed, thisdiscussion does not only apply to numbers: for instance, an algebraic or trigono-metric expression in three variables can be viewed as a function whose domainis some subset of three-dimensional space, and it is both natural and straightfor-ward to extend the ideas of limit and continuity so as to apply also in this arena.These remarks add force and breadth to the point we were making in the secondparagraph of this chapter: there is much to be gained, in terms of eciency anduniversality and understanding, if we can formulate a denition of limit whichencompasses all the special cases that turn up in elementary analysis, and whichcan apply equally well to real numbers, complex numbers, points in space or anyother similar class of objects. The subsidiary question raised by the last sentenceis: what does similar mean in this context? What exactly is it about R,C and R3

that permits such a discussion to take place?Now, most of what we have said so far is an introduction, not to something

rich and strange, but to something that you have already studied.What it is aboutR,C and R3 is simply the fact that they are metric spaces. You will know from

SETS AND MAPPINGS 3

your study of these structures that as soon as we can spell out, in some reason-able fashion, what we mean by distances between objects in a set, then limits ofsequences within that set make sense, and so does continuity of a function fromthat set to itself or to any other similarly structured set, and large swathes of elem-entary analysis transfer themselves painlessly to the new, wider context: that is,the same results are valid, and they are provable by the same methods as before.This is the origin and, in part, the purpose of metric space theory. (We shallbriey review it later in the present chapter.)

Yet there are circumstances in which limiting processes and continuous trans-formations make sense, but where the metric-space notion is inadequate todescribe them. This is a central theme of topology, but one which it may be dif-cult to illustrate well on the basis of your previous study. Perhaps the followingremarks will be sucient for the moment. It is explicit in the formal denitionof a metric space that (i) the distances from x to y and from y to x are equal; (ii)objects at zero distance are identical; and (iii) distances are always non-negativereal numbers. However, any driver in a city with one-way streets knows that thereare real-world situations which cannot be modelled using (i) as an axiom. Again,a useful measure of distance between continuous functions on a bounded closedinterval [a, b] is

d( f , g) = ba

| f (x) g(x)| dx ;

yet some discontinuous functions can also be integrated, and one then nds thatfunctions f and g that dier at only a nite number of values of x have d(f , g) = 0,although it would not then be true to say that f = g, despite the aspirations of (ii).Lastly, unbounded functions can often be approximated by using bounded ones,but a thoughtless and slavish use of metric concepts to express this approximationcan give rise to innite distance assignments, contrary to (iii). Our overall pointis that the idea of a metric space is just not subtle or general enough to providea full and reliable description of every context in which limits and continuity areapplicable. This is the task that general topology takes on. We shall exhibit a rangeof instances where it successfully accomplishes its mission.

Sets and mappings

We assume familiarity with very basic material on sets, such as:

1.1 Proposition (De Morgans laws) For any family {Bi : i g I} of subsets of agiven set X,

(i) X \}igIBi =|igI(X \Bi),(ii) X \|igIBi =}igI(X \Bi), and1.2 Proposition (the distributive laws) For any subsetA of a given set X and anyfamily {Bi : i g I} of subsets of X,

4 1 INTRODUCTION

(i) A|}igI

Bi =}igI

(A| Bi),

(ii) A}|igI

Bi =|igI

(A} Bi).

Also assumed are the elementary ideas of mappings (which we feel free to callmaps or functions without any implied change of emphasis) such as domain,codomain, range, one-to-one/11/injective, onto/surjective, bijective and inverseswhen they exist; note in particular, for a given map f : X C Y , the notations

f (A) = {f (a) : a g A},

f 1(B) = {x g X : f (x) g B}

for each subset A of X and each subset B of Y . The notation f 1(B) is perhapsunfortunate, but in widespread use: it might tempt you to think that f has aninverse mapping, but this is not generally the case (indeed, it is expressly not thecase unless f is both 11 and onto). The essential thing to remember about the setsf (A) Y and f 1(B) X is which points are in them:

y g f (A)K Mx g A such that f (x) = y,

x g f 1(B)K f (x) g B.

As processes on sets, f and f 1 behave rather nicely; we recall the following easybut vital rules.

1.3 Proposition

(a) Let f : X Y and let {Ai : i g I} be a family of subsets of X. Then:(i) f (}{Ai : i g I}) =}{f (Ai) : i g I},(ii) f (|{Ai : i g I}) |{f (Ai) : i g I},(iii) equality does not usually obtain in (ii).

(b) Let f : X Y and let {Bi : i g I} be a family of subsets of Y . Then:(i) f 1(}{Bi : i g I}) =}{f 1(Bi) : i g I},(ii) f 1(|{Bi : i g I}) =|{f 1(Bi) : i g I},and also(iii) f 1(Y \B) = X \ f 1(B) for any B Y .

(c) Let f : X Y and A X,B Y . Then:(i) f (f 1(B)) B,(ii) A f 1(f (A)),(iii) equality obtains in (i) for all B if and only if f is onto,(iv) equality obtains in (ii) for all A if and only if f is 11.

On the infrequent occasions when a function has to be described without need-ing to acquire a referring symbol, we use the x C y notation. For example, a

ZORNS LEMMA 5

passing reference to the general quadratic (real) function can be made by writingthe map x C ax2 + bx + c from R to R.

We assume that you are condent about the distinction between nite sets andinnite sets. (Note in particular that the empty set counts as nite.) We call a setcountably innite if there is a bijection between it and the setN (less formally, if itselements can be listed as the terms of an endless list, an (innite) sequence). By acountable set we intend one that is either nite or countably innite.Uncountablemeans not countable. Keep in mind that Q is countable, that any interval in R isuncountable (with the obvious exception of degenerate one-element intervals ofthe form [a, a]), that the union of countably many countable sets is countable andthat the (Cartesian) product of a nite number of countable sets is countable (weshall review the idea of product set in Chapter 5).

Be clear about the dierence between element and subset, especially when deal-ing with a set of sets. For instance, if > is the collection of all open circular discsin the coordinate plane, and ? is the set of open circular discs centred on the ori-gin and of rational radius, and D is the particular disc B(0, 1) whose centre is theorigin and whose radius is 1, then D is an element of ? but not a subset, ? is asubset of > but not an element, and so on. These distinctions may seem petty (andacademic in the derogatory sense!) but they really do matter. Be careful, therefore,always to use the symbols andg correctly.

Zorns lemma

The one somewhat sophisticated idea from set theory that we very occasionallyneed to use is Zorns lemma, and we shall set out the lengthy but easy details of itsbackground here.

Recall that, if X is a set and is a relation on X, then:

is a partial order if it is reexive, anti-symmetric and transitive(that is, x x L x g X,

(x y and y x)I x = y,(x y and y z)I x z);

is a total order if, in addition to being a partial order, it satises the conditionthat every two elements of X are comparable; that is,

given x, y g X, either x y or y x; is awell-ordering if, in addition to being a total order, it satises the condition:

every non-empty subset of X has a least member(that is, = A X implies M a0 g Asuch that, for every a g A, a0 a);

whenever x y and x y, it is customary to write x < y. The notations x < yand y > x are interchangeable, as are the notations x y and y x.

6 1 INTRODUCTION

By way of illustration, set inclusion is a partial order on any collection of setsbecause the three necessary rules

A A always,(A B and B A)I A = B,(A B and B C)I A C

are obviously satised. Again, on the setN the relation div dened by

x div yK x is a factor of y

is a partial order because

x div x always,(x div y and y div x)I x = y,(x div y and y div z)I x div z

evidently hold. Neither of these is (generally) a total order: for example, neither6 div 10 nor 10 div 6 is true, and neither {1, 2} {2, 3} nor {2, 3} {1, 2} is true.Furthermore, the usual order onN orQ orR is obviously a total order, and theusual order on N is even a well-ordering, but the usual order on Q is not a well-ordering (what is the least of the positive rationals whose squares exceed 2? thereisnt one!) and the usual order on R is not a well-ordering (there is, for instance,no least member of the open interval (0, 1)). Note also that a set which has a partialorder is called a poset, and that a set with a total order is called a chain. A set witha well-ordering is just called a well-ordered set.

Any non-empty subset A of a poset (X,) can be regarded as a poset in its ownright if we merely restrict the ordering on X to apply only to the elements of A. IfX is a poset but is not totally ordered, there will certainly be subsets of X which aretotally ordered. These are called chains in X. For instance, {3, 6, 30, 120, 1200} is achain in (N, div) and {{a}, {a, b}, {a, b, c, d}, {a, b, c, d, e, f }} is a chain in the subsetsof the alphabet under set inclusion.

If A is a subset of a poset (X,) and y is an element of X, we call y an upperbound of A if, for every a g A, a y.

If A is a subset of a poset (X,) and y is an element of A (note: of A this time!),we say that y is a maximal element of A if (a g A, y a) together imply thata = y: that is, if no element of A is strictly greater than y. This is not the sameas greatest element: although the greatest element of A (if it exists) will certainlybe a (unique) maximal element of A, sets can readily have many dierent max-imal elements (none of which could therefore be a greatest element for that set).For example, when {3, 4, 5, 6, 7, 8, 14} is ordered by div, 5 and 6 and 8 and 14 aremaximal elements for it.

Now that the undergrowth has been cleared, we can at last state Zorns lemma.

1.4 Zorns lemma If, in a poset, every chain has an upper bound, then the posethas at least one maximal element.

ZORNS LEMMA 7

The proof of this assertion lies outside the scope of the present text, but we shalldemonstrate here a small example of how to use the lemma.

1.5 Exercise Show that there is, in the coordinate plane R2, a subset D with thefollowing two properties:

(1) no three points of D are collinear, and(2) every point of R2 \D lies on the line through two points of D.

Solution Let us call B R2 line-shy if no three of its points are collinear. Let Lbe the collection of all line-shy subsets. Then (L,) is a poset. We seek to applyZorns lemma to it.

Let T be any chain in (L,), that is, T is a family {Ci : i g I} of line-shy setsevery two of which are comparable (under). What can we say about their unionK = }{Ci : i g I}? Any three elements p, q, r of K will need to belong to threesets Ci(p),Ci(q),Ci(r) within T, and the comparability clause says that one of thosethree sets let us call it Cj contains the other two. So p and q and r all lie in theline-shy set Cj, and cannot therefore be collinear. Thus, K is in L, and is an upperbound in (L,) for the entire chain T.

Now that we have seen that Zorns lemma can be applied to this poset, it assuresus that L has a maximal elementD. SinceD is an element of L, (1) holds. Since it ismaximal, any attempt to augment D even by the adjunction of one extra elementwill push it outside L. That is, for any s g R2 \D,D} {s} does not belong to L, andthree of its points must therefore be collinear. The three points cannot all belongto D (since D is line-shy), so s is one of the three. This proves (2).

Actually, the above proof is a classic case of the use of a sledgehammer to cracka walnut, because a circular or elliptical contour would do the job of the set D asdescribed in 1.5 without any appeal to Zorn! Nevertheless, our new sledgehammerwill crack tougher nuts than this: for instance, almost exactly the same argumentwill produce a subset of the plane satisfying (1) and (2) that contains any givenline-shy subset, or will let us carry out the whole exercise within any given subset ofthe plane. More importantly, this argument shows how, in practice, Zorn is usuallyapplied, and we shall use it later in at least one scenario where no quick and easysolution is available.

Zorns lemma is closely associated with another major and sophisticated resultin set theory which it may be useful to mention at this stage:

1.6 The well-ordering theorem Let X be any set. Amongst all the relations onX, there is at least one that is a well-ordering: that is, any set can be well-ordered.

(The demonstration that 1.6 may be deduced from 1.4 also lies well outside thescope of the present text. We emphasise that, if the given set X happens to comewith a (partial or total) order already in place, then the well-ordering guaranteedhere may well be utterly unrelated to it.)

8 1 INTRODUCTION

The least uncountable ordinal

One of the immediate consequences of 1.6 is that there exist uncountable well-ordered sets. For instance, we could just impose a well-ordering on R, and it isthen a slight additional technicality to arrange that, under this well-ordering, theset shall have a greatest element: let us call it m for the moment. Under this well-ordering (which, we should again emphasise, will have no relationship at all withthe natural ordering of the real numbers), R will possess a least element, con-ventionally denoted by 0. We may use the familiar notation of intervals in thiswell-ordered set so that, for example, [0,m] is the whole ofR, and we see that thereexist values of x g R for which the so-called initial segment [0, x) is uncountable.The well-ordering now assures us that there is a least such x: so, denoting it by 81,we have that [0,81) is uncountable but, for every t < 81, the initial segment [0, t)is countable.

The well-ordered set [0,81) we have just been describing is of considerable im-portance in topology (and certainly elsewhere in mathematics) and it is essentiallyunique in the sense that, if W1 and W2 are two well-ordered sets that are un-countable and have all of their initial segments countable, then there is a bijectionf : W1 CW2 that preserves the ordering (that is to say, x < y inW1 if and only iff (x) < f (y) inW2). It is called, for obvious reasons, the least uncountable ordinal.From our point of view, the most important of its properties is the following:

1.7 Proposition Any countable subset of the least uncountable ordinal has anupper bound.

Let us bemore specic as to why this result interests us in the present context. If, inthe notation above, C [0,81) and C is countable, then by 1.7 there exists t < 81such that for every c g C we have c t. That means that the interval (t,81)contains none of the elements of C. In eect, if we now restore 81 as top elementto our ordered set and choose to work in [0,81] rather than in [0,81), no sequencein [0,81) can get close to 81, because some neighbourhood (t,81] screens 81 ofrom all the terms in the sequence: for the range of the sequence is, of course,a countable set. We are, of course, getting too far ahead of our exposition now,for we have not yet dened neighbourhood of a point nor limit of a sequence inways that are meaningful in such a context as this; but now we shall move towardsremedying this defect through a brief revision of the elements of metric spacetheory that the reader of such a text as this is expected to have encountered already.

Metric spaces

A metric on a non-empty set is an assignment of a real distance to each pair ofelements of the set that mirrors, in the following sense, how physical distancesbehave:

METRIC SPACES 9

1.8 Definition Ametric d on a non-empty setM is a mapping fromMM intothe real line R such that, for all x, y, z inM:

(i) d(x, y) 0,(ii) d(x, y) = d(y, x),(iii) d(x, y) = 0 if and only if x = y and(iv) d(x, z) d(x, y) + d(y, z).

Then the pair of objects (M, d) is called ametric space.

1.9 Examples

(i) Any non-empty subset of the real line or the coordinate plane or ordinarythree-dimensional space or, indeed, of ordinary n-dimensional spacebecomes a metric space when we choose to measure distance in the familiarEuclidean way. Furthermore, there are non-Euclidean metrics even on sucheveryday spaces as these: for instance, both

d ((x1, y1), (x2, y2)) = |x1 x2| + |y1 y2|

and

d ((x1, y1), (x2, y2)) = Max{|x1 x2|, |y1 y2|}

also dene useful and natural metrics d , d on the coordinate plane.(ii) Any non-empty set at all may be made into a so-called discretemetric space

by declaring the distance between every two distinct points to be 1.(iii) The family of continuous real-valued functions on [0, 1] becomes a metric

space when we dene the function-to-function distance asd(f , g) =

10 | f (x) g(x)| dx , and it becomes a signicantly dierent

metric space when, instead, we dene distance asD(f , g) = Max{| f (x) g(x)| : 0 x 1}.

(iv) In contrast to what usually happens in algebraic areas, where it is onlycertain special subsets of a structure (subgroups, subelds, sub-vectorspaces and so on) that can be made into a substructure under the sameprocesses, absolutely any non-empty subset A of a metric space (M, d) canbe made into a sub-metric space merely by restricting the metric d to applyonly to pairs of points from A. When this is done, A is called simply asubspace of (M, d). This will generate a wide variety of metric spacesstarting o from almost any standard example, as we have already indicatedin (i) above. By way of illustrating how wide the divergence in behaviour ofa subspace from its parent space can be, we now describe an example whichwill prove useful at several points later in the text.

10 1 INTRODUCTION

1.10 Example: the Cantor excluded-middle sets Whenever A is a set of realnumbers consisting of the union of nitely many pairwise disjoint bounded closedintervals

A = [a1, b1]} [a2, b2]} [a3, b3]} } [an, bn],let us dene the set A to be the union of the top and bottom one-tenths of eachof these intervals:

A =[a1, a1 +

110

(b1 a1)]}[a1 +

910

(b1 a1), b1]}

}[an +

910

(bn an), bn].

Since the set A is itself the union of a similar family of intervals, we may apply to it in its turn to create A and, further, A and so on.

The case which most directly concerns us is that in which we begin this processwith [0, 1], because standard decimal notation then gives us a simple way to deter-mine exactly which numbers belong to [0, 1] and all the later iterates. In [0, 1],we nd exactly those numbers that can be given decimal expansions with only 0 or9 in the rst decimal place. This is because, in removing the central eight-tenthsof the interval, we have taken out those whose rst decimal digit was 1, 2, 3, or 8. Notice the importance of the word can: for instance, 110 is usually expressedas the decimal 0.1, but it can be written as 0.09, which agrees with its inclusionin [0, 1]. For the same reason, [0, 1] comprises those real numbers that can begiven a decimal expansion with only 0 or 9 in the rst two places, [0, 1] com-prises those whose expansions can be done with only 0 or 9 in the rst three places,and so on.

Having thus generated an innite nested sequence

[0, 1] [0, 1] [0, 1] [0, 1] n[0, 1] of sets on the real line, it is natural to ask: what is their intersection? (Let us denoteit by Cantor10.) Arithmetically, the answer is obvious from the discussion above:Cantor10 comprises exactly those real numbers whose entire decimal expansionscan be expressed using only 0s and 9s. The detailed structure of the metric spaceCantor10 is, however, less obvious and more interesting, as we shall see.

The bio-historical accident that humans usually do their arithmetic to base 10has no essential role to play in the above construction: we could have opted towork to any other base b greater than 2 without signicantly aecting the struc-ture of the resulting space (let us denote it by Cantorb), as we shall make clearin Chapter 3. Had we chosen, for example, to count to base 3, we should haveremoved central thirds from the various intervals instead of central eight-tenths,and tresimal expansion of real numbers then gives us a convenient way to keeptrack of what has been excised and what ultimately remains: because now, [0, 1]

METRIC SPACES 11

comprises the real numbers whose tresimal representation can be done withoutusing 1 as the rst digit, [0, 1] comprises those whose representations can avoidusing 1 as the rst or second digit, and so on, while Cantor3 itself consists of allreal numbers that are expressible in the form

0.a1a2a3a4 [3] = an

3n,

where each base-3 digit an is either 0 or 2. It is this version of the Cantor set that ismost commonly presented, and from which its customary name of middle thirdsset is derived.

1.11 Exercise

(i) Show that the average of any two distinct elements of Cantor5 lies outsideCantor5.

(ii) Verify that the corresponding statement for Cantor3 is false.

1.12 Definition Given a metric space (M, d):

(i) an open ball is a subset of the form B(x, r) = {y g M : d(x, y) < r} for somex g M and real r > 0;

(ii) an open set is a set that can be expressed as a union (nite or innite) ofopen balls;

(iii) a closed set is a subset ofM whose complement is an open set.

It is evident that M itself and the empty set are open and that the union of anarbitrary family of open sets is open, and it is an easy induction argument to checkthat the intersection of any nite family of open sets is open. Be aware that a setcan be both open and closed (for instance, in a discrete space, every set is) and thatmany (most?) sets are neither open nor closed.

To bring out the connections with the historical roots of this material, we canopt to dene continuity and sequential limits for metric spaces exactly as one doesin elementary analysis, thus:

1.13 Definitions

(i) Given a pair of metric spaces (M, d) and (M, d ) and a mapping f : M C M,we say that f is continuous if, for each x g M and each positive real number%, there is a positive real number $ such that

y g M, d(x, y) < $ together imply d (f (x), f (y)) < %.

(ii) Given a metric space (M, d), a sequence (xn)ngN inM and an element x ofM, we say that (xn)ngN converges to the limit x (and we write xn C x) if, foreach positive real number %, there is a positive integer n0 such that

n g N, n n0 together imply d(xn, x) < %.

12 1 INTRODUCTION

However, in practice it is often more convenient to redene ideas such as thesein terms not of distance but of open sets and/or open balls. (Such characterisationstend to be easier to handle, in part because the fundamental behaviour of open setsis simpler than that of distance measurements.) For example:

1.14 Lemmas

(i) Given a pair of metric spaces (M, d) and (M, d ) and a mapping f : M C M,then f is continuous if and only if:

f 1(G) is open in (M, d) whenever G is open in (M, d ).

(ii) Given a metric space (M, d), a sequence (xn)ngN inM and an element x ofM, then (xn)ngN converges to x if and only if:

given open set G such that x g G, we have xn g G for all suciently large n.

(Incidentally, in (ii) above we may replace open set by open ball or by open ballcentred on x and the result remains valid.)

Continuing for amoment with the theme of equivalent denitions, we point outthat sequential convergence in its turn will serve to provide alternative denitionsfor many of the ideas in metric spaces; witness the following few results:

1.15 Lemmas

(i) Given a pair of metric spaces (M, d) and (M, d ) and a mapping f : M C M,then f is continuous if and only if:

whenever a sequence xn C x in (M, d), then f (xn)C f (x) in (M, d ).

(ii) Given a metric space (M, d) and a subset A ofM, then A is a closed set if andonly if:

whenever a sequence (xn)ngN of elements of Aconverges to x in (M, d), then x g A.

1.16 Lemma We consider the following statements about a non-empty subset Aof a metric space (M, d):

(a) A is closed and bounded;(b) within every family of open sets whose union contains A, there is a nite

subfamily whose union already contains A;(c) every sequence in A has a subsequence that converges to an element of A.

Then:

(i) in all cases, (b) and (c) are equivalent, and they imply (a);(ii) in the Euclidean spaces, all three are equivalent.

EXPANSION OF CHAPTER 1 13

Proofs of these results will be found in any book that deals with the elementarytheory of metric spaces, so we shall not include them here. Presently our mainpurpose apart from reminding the reader of material almost surely encounteredearlier, andwith a view to accessing easy illustrative examples is to ag up the factthat once we move into topology proper and lose the metric, the various equiva-lences that are so much a feature of metric space theory and practice can no longerbe relied upon. In some cases (1.15, for instance), what was in the metric setting anecessary and sucient condition will turn out to be only necessary in topology;in others (1.16s (b) if and only if (c) illustrates this), it will not even be that; inothers still (consider, for example, (b) implies (a) in 1.16), it may lose meaningaltogether as soon as there is no longer a metric.

Expansion of Chapter 1

1.1(i) x g RHS K (Li g I) x g X \Bi

K (Li g I) x g Bi

K x /g}igI

Bi

K x g X \}igI

Bi = LHS.

(ii) x g LHS K x g|igI

Bi

K Mi g I such that x g BiK Mi g I such that x g X \BiK x g}

igI(X \Bi) = RHS.

1.2(i) x g LHS K x g A and Mi g I such that x g Bi

K Mi g I such that x g A| Bi

K x g}igI

(A| Bi) = RHS.

(ii) Similar.

14 1 INTRODUCTION

1.3

(a) (i) (For y g Y :)y g LHS K Mx g}

igIAi such that y = f (x)

K Mi g I, x g Ai such that y = f (x)K Mi g I such that y g f (Ai)K y g}

igIf (Ai) = RHS.

(ii) y g LHS K Mx g|igI

Ai such that y = f (x)K Mx such that,Li g I, x g Ai and y = f (x)I (Li g I) y g f (Ai)K y g|

igIf (Ai) = RHS.

(iii) For instance, dene f : R R by f (x) = x2, take A1 = [2, 0],A2 = [0, 3].Then f (A1)| f (A2) = [0, 4]| [0, 9] = [0, 4],

but f (A1 | A2) = f ({0}) = {0}.

(b) (i) x g LHS K f (x) g}igI

Bi

K Mi g I such that f (x) g BiK Mi g I such that x g f 1(Bi)K x g}

igIf 1(Bi) = RHS.

(ii) Similar.

(iii) x g LHS K f (x) g Y \BK f (x) g BK x g f 1(B)K x g X \ f 1(B) = RHS.

(c) (i) y g LHS I Mx g f 1(B) such that y = f (x)I y g B = RHS.

(ii) x g A I f (x) g f (A)I x g f 1(f (A)) = RHS.

(iii) (A) Suppose f is onto.For any y g B, Mx g X such that y = f (x)


and then x g f 1(B),so y g f (f 1(B)).This shows B f (f 1(B)),and now (i) shows we have equality of these sets.

(B) Suppose f is not onto.Then My g Y \ f (X).Put B = {y}. Then f (f 1(B)) = f () = = B.

(iv) (A) Suppose f is 11.For any x g f 1(f (A)), f (x) g f (A),so Mx g A such that f (x) = f (x).But (since f is 11) x = x,so x g A.This shows f 1(f (A)) A,and now (ii) shows we have equality of these sets.

(B) Suppose f is not 11.Choose x1 = x2 such that f (x1) = f (x2).Put A = {x1}. Then x2 g f 1(f (A)),so A = f 1(f (A)).

1.7 Suppose, if possible, that:C [0,81),C is countable and C is not bounded above in [0,81).

For each x g [0,81), x is not an upper bound of C,so x < tx for some tx g C.That is, [0,81) }tgC[0, t).But each [0, t) (where t < 81) is countable,so }tgC[0, t) is a countable union of countable sets, and thereforecountable.This is a contradiction, since [0,81) is uncountable.

1.11 (i) Using arithmetic in base 5, the typical element of Cantor5 can berepresented as a pentimal 0.a1a2a3a4 [5], where each base-5 digitai is either 0 or 4.The average of two of these (a and b, say) will therefore take the form0.m1m2m3m4 , where eachmi is either 0 or 2 or 4.If a and b are distinct, one of them will have a 0 in a pentimal placewhere the other has a 4, so at least onemi will equal 2.

16 1 INTRODUCTION

Then it will not be possible to represent (a + b)/2 as a pentimal usingonly 0s and 4s as digits,so (a + b)/2 does not belong to Cantor5.

(ii) If we try the same argument in base 3 to describe Cantor3, then 0 and 2are the permitted digits and the conclusion reached is that at least onedigitmi of the average of a and b equals 1.But it may still be possible to represent that average without using thedigit 1 because, for instance, 0.02201 [3] and 0.022002 [3] are the samebase-3 number;hence the argument of (i) above will fail.More explicitly, 0 and 2/3 belong to Cantor3 (in tresimal form they are0.0 and 0.2) but so is their average 1/3, since it can be expressed notonly as 0.1 but also as 0.02.

2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Topological spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some elementary concepts

2.1 Definition Let X be a non-empty set. A topology on X is a collection 3 ofsubsets of X such that

(i) X and belong to 3,(ii) the union of any set of members of 3 is a member of 3,(iii) the intersection of any nite set of members of 3 is a member of 3.

The pair of objects (X, 3) is called a topological space. We often write it simply asX, provided that it is obvious which topology we have in mind. The members of3 are called the open sets, or the 3-open sets. Due, perhaps, to the geometrical andmetric-space roots of the subject, the elements of X are often referred to as thepoints of the space.

2.2 Examples

(i) For any metric space (M, d), we know that its open sets do form a topologyonM, which we shall usually write as 3d. Any topology that arises like this issaid to bemetrisable. In particular, the usual metrics on Rn (for each n 1)and on its subsets give rise to topologies on R and its powers and theirvarious subsets, which we refer to as the usual topologies on these sets. Thenotation 3usual will be used to denote these most familiar topologies.

(ii) On any set X, {X,} is a (not very interesting!) topology, called the trivialtopology, which we sometimes denote by 3triv.

(iii) At the other extreme, any set X supports a so-called discrete topology 3disc,dened to be the entire powerset P(X) (that is, the set of all subsets of X).This happens to be metrisable: the metric which assigns a distance of 1 toeach two distinct elements of X produces it.

(iv) If X is innite, then a subset A of X is called conite when its complementX \A is nite. The collection

3cf = {G X : G is conite or G = }

is a topology on X, the conite topology on this set.

18 2 TOPOLOGICAL SPACES

(v) If X is uncountable, then a subset A of X is called cocountable when itscomplement X \A is countable. The collection

3cc = {G X : G is cocountable or G = }

is a topology on X, the cocountable topology on this set. (You should checkthat it is a topology.)

(vi) If X is any set and p is an arbitrary point of X, then the collection

)p = {G X : p g G or G = }

is a topology on X, the included-point topology (based at p) there.(vii) If X is any set and p is an arbitrary point of X, then the collection

p = {G X : p g X \G or G = X}

is a topology on X, the excluded-point topology (based at p) there.

We shall feel free to refer to a discrete space, a trivial space, a conite spaceand so on, when the topology in play is so named. Needless to say, there aremany examples of much more interesting topologies than these, and we shall seesome of them. However, the above handful of basic examples will turn out to besurprisingly useful.

2.3 Definition Let (X, 3) be a (topological) space, p an element of X and N asubset of X. We call N a neighbourhood of p (or, when we need to be fussy, a3-neighbourhood of p) if there is an open set G in 3 for which

p g G N

(Fig. 2.1)

p

NG

Fig. 2.1 N is a 3-neighbourhood of p (see 2.3).

SOME ELEMENTARY CONCEPTS 19

2.4 Exercise Determine how to recognise neighbourhoods of points in thespaces of 2.2. Keep in mind that neighbourhoods do not have to be open setsthemselves.

2.5 Lemma The intersection of nitely many neighbourhoods of a point p (inany topological space) is a neighbourhood of p.

2.6 Lemma Given a space (X, 3) and any subset A of X, we have that A g 3 ifand only if A is a 3-neighbourhood of every point in A.

2.7 Definition A subset F of a space (X, 3) is called closed (or 3-closed) when itscomplement X \ F is open.

2.8 Warning Some subsets are both open and closed. Many subsets are neitheropen nor closed. Therefore:

to prove that a set is open, it is not enough to show that it cannot be closed; to prove that a set is closed, it is not enough to show that it cannot be open; to prove that a set is not open, it is not enough to show that it is closed; to prove that a set is not closed, it is not enough to show that it is open.

This paragraph highlights the most common source of mistakes in elementarytopology!

Because of how we have dened closed, it is hardly surprising that the basicproperties of the closed sets are mirror images of those of the open sets:

2.9 Proposition In any topological space (X, 3):

(i) X and are 3-closed,(ii) the intersection of any set of closed sets is a closed set,(iii) the union of any nite set of closed sets is a closed set.

2.10 Definition Given a subset A of a space (X, 3), the closure of A, usually writ-ten A or, when we really have to, A 3 , is the intersection of all the 3-closed sets thathappen to contain A. It is closed. It is the smallest closed set that does contain A.The set A is closed if and only if A = A.

Here are the basic properties of the closure process.

2.11 Proposition For any subsets A, B of a space (X, 3):

(i) = ,(ii) A A always,

(iii) A = A always,(iv) A} B = A} B.


2.12 Warning Note that 2.11(iv) extends easily to the union of any nite numberof sets, but that it does not extend to unions of innitely many.

2.13 Exercise Explore examples of closure in some spaces. For instance, deter-mine the closure of an arbitrary set in a conite space, or in an included-pointspace. Note in particular that, in (R, 3usual),Q = R, whereQ is the set of rationalnumbers, and that the Cantor excluded-middle set (using any arithmetical base)is closed.

Here is a highly useful little result tying together closure and neighbourhoods inany given space.

2.14 Lemma For A X and p g X, we have that p g A if and only if everyneighbourhood of pmeets A (Fig. 2.2).

p

A

N

Fig. 2.2 All neighbourhoods of pmeet A (see 2.14).

Subspaces

2.15 Definition Let A be a non-empty subset of X, where (X, 3) is a given space.The set of traces onto A of the 3-open sets by which we mean the collection

{A | G : G g 3}

is a topology on A, sometimes written as 3A or as 3|A. It is called the topologyinduced on A by 3, or, more usually, the subspace topology. The space (A, 3A) istermed a subspace of (X, 3) (Fig. 2.3).

2.16 Lemma In the above notation, the 3A-closed sets are precisely those of theform A| F, where F is 3-closed.

SUBSPACES 21

A

A G

XG

Fig. 2.3 A as a subspace of (X, 3) (see 2.15).

2.17 Lemma In the above notation, given p g A, the 3A-neighbourhoods ofp are precisely the sets of the form A | N, where N is a 3-neighbourhood of p(Fig. 2.4).

A

N

p

G

Fig. 2.4 Neighbourhoods in the subspace topology (see 2.17).

2.18 Lemma In the above notation, given B A, the 3A-closure of B is

B 3A = A| B 3 .

2.19 Warning Although 2.162.18 suggest (correctly) that subspaces are usuallyeasy to handle because the structure just gets traced or shadowed onto the subsetthat carries the subspace topology, there is also a rich source of errors here. Sup-pose that we are given (X, 3) and B A X. If we say B is open, then there aretwo distinct things that we might mean: that B is a 3-open subset of X which hap-pens to be contained in A, or that B is a 3A-open set. It is vital to realise that theseare dierent; similar remarks apply to closed, to neighbourhood, to closure andso on. (You should be able to nd easy illustrations of these comments in the realline, using sets no more complicated than intervals.)


A way to prevent this diculty arising is consistently to use terms like 3-open,3A-open, 3-neighbourhood, 3A-closure and so on whenever there is a genuine riskof ambiguity. We note two small positive results also:

2.20 Lemma If A is itself 3-open, then any 3A-open set will be also 3-open; if Ais itself 3-closed, then any 3A-closed set will be also 3-closed.

2.21 Definition A property of topological spaces is called hereditary if, whenevera space possesses the property, then so must all of its subspaces. An example ismetrisability.

Many properties are inherited not by all subspaces but by important special classesof subspaces, and it is useful to extend the denition. A property that is inheritedby every subspace (A, 3A) for whichA is 3-closed is called a closed-hereditary prop-erty. A property that is inherited by every subspace (A, 3A) for which A is 3-openis called an open-hereditary property.

2.22 Examples The property every non-empty open set is uncountable is nothereditary, but it is open-hereditary. The property every countable set is closedis hereditary, and therefore also closed-hereditary and open-hereditary. Othersomewhat contrived examples like these can be devised easily enough but, onceagain, we shall see better and more useful examples later.

Exercises Essential Exercises 413, 15 and 17 are based on the material in thischapter. It is particularly recommended that you should try numbers 4, 5, 6, 7, 8,11 and 13.


2.4

(i) N is a neighbourhood of xK M% > 0 so that B(x, %) N.(ii) N is a neighbourhood of xK N = X.(iii) N is a neighbourhood of xK x g N.(iv) N is a neighbourhood of xK x g N and X \N is nite.(v) N is a neighbourhood of xK x g N and X \N is countable.(vi) N is a neighbourhood of xK p g N and x g N.(vii) N is a neighbourhood of xK

x = p & N = Xorx = p & either (x g N and p g N) or N = X.


2.5 Let N1,N2, . . . ,Nj be nitely many neighbourhoods of p (in X).Choose open sets G1,G2, . . . ,Gj so that

p g G1 N1, p g G2 N2, . . . , p g Gj Nj.

Thenj

|1

Gi is open, and p gj

|1Gi

j

|1Ni

thereforej

|1Ni is a neighbourhood of p.

2.6(i) Suppose A is 3-open in (X, 3).

For each x g A, we choose G = A g 3 and we have x g G A.Therefore A is a neighbourhood of x.That is, A is a neighbourhood of every point in A.

(ii) Suppose A is a neighbourhood of each of its own points. So:for each x g A we can choose Gx g 3 such that x g Gx A.Now}xgAGx A obviously. But also each point x of A belongs to Gx, soA }xgAGx.Thus A =}xgAGx, a union of 3-open sets, therefore 3-open itself.

2.9(i) X and are closed because their complements ( and X) are open.(ii) If {Fi : i g I} is any family of closed sets, then each X \ Fi is open, so}igI(X \ Fi) is open.

But X \|igI

Fi =}igI

(X \ Fi)

therefore |i

Fi is closed.

(iii) If F1, F2, . . . , Fn are nitely many closed sets then

X \n

}1

Fi =n

|1

(X \ Fi)

= a nite intersection of open setstherefore open,

son

}1

Fi is closed.


2.11

(iv) A1 } A2 is closed and contains A1 } A2

therefore A1 } A2 A1 } A2. (1)But also each Ai A1 } A2,

therefore Ai A1 } A2therefore A1 } A2 A1 } A2. (2)

By (1) and (2) we have

A1 } A2 = A1 } A2.

2.12 A1 } A2 } . . .} An is closed and contains A1 } A2 } . . .} An.

Therefore A1 } A2 } . . .} An A1 } A2 } . . .} An. (1)But also each Ai A1 } . . .} An

therefore Ai A1 } . . .} An

thereforen

}1

Ai A1 } . . .} An. (2)

By (1) and (2) we have

n

}1

Ai =n

}1

Ai.

In (R, 3usual):

}1

(1

n + 1, 1)= (0, 1) = [0, 1],

but}1

(1

n + 1, 1)=

}1

[1

n + 1, 1]= (0, 1].

2.13 If the complement in R of the closure of Q were non-empty, it would beopen and would contain an open interval (a, b). This interval would containno rational numbers, contradicting the well-known density of the rationalsin the reals.


In the real line, each bounded closed interval [a, b] is a closed set because itscomplement is the union of two (unbounded) open intervals, so the sets bywhich we recursively constructed the Cantor set nite unions of boundedclosed intervals are closed. The Cantor set was dened as the intersectionof these, and is thus closed also.

2.14

(i) Let p g A.Suppose that N is any neighbourhood of p.If N does not meet A (that is, their intersection is empty) then choose G g 3such that

p g G N

and observe that G does not meet A either,

that is, X \G A, where X \G is closed,therefore X \G A.

Now p g A tells us p g X \G, that is, p g G, contradiction!So N has to intersect A.

(ii) Let p g A.Then X \A is an open set including pand therefore is a neighbourhood of pand it certainly cannot intersect A!So not every neighbourhood of pmeets A.

2.16 Let F1 A X.(i) Suppose F1 is 3A-closed

that is, A \ F1 is 3A-openthat is, A \ F1 = A| G for some G g 3.But then F1 = A| (X \G)and X \G is 3-closed.

(ii) Suppose F1 takes the form A| F, where F is 3-closed.Then X \ F is 3-openand A| (X \ F) = A \ F1 is 3A-open.Therefore F1 is 3A-closed.


2.17 Let p g A X,N1 A.(i) Suppose N1 is a 3A-neighbourhood of p.

Then MG1 g 3A such that p g G1 N1and G1 takes the form A| G, some G g 3.Now p g G N1 } Gso N1 } G is a 3-neighbourhood of pand

A| (N1 } G) = (A| N1)} (A| G) = N1 } G1 = N1,

so N1 is of the form A | a 3-neighbourhood of p.

(ii) Suppose N1 = A| N, where N is a 3-neighbourhood of p,that is, p g G N for some G g 3.Then p g A| G A| N,that is, p g A| G N1,where A| G is 3A-open.Therefore N1 is a 3A-neighbourhood of p.

2.18 Let B A X.Now A| B3 is 3A-closed (see 2.16)and contains A| B = B.

Therefore B3A A| B3 . (1)

On the other hand, any 3A-closed set that contains B takes the form A| F,where F is 3-closed

and B Ftherefore B3 F

therefore A| B3 A| F.

Yet B3A is the intersection of all such sets A| F,

therefore A| B3 B3A . (2)

Now we combine (1) and (2).


2.19 Let X = (R, 3usual),A = (0, 1].Then (0, 12 ] is closed in the subspace A(since it equals A| [0, 1])

but not closed in X.Also, ( 12 , 1] is open in the subspace A

(since it equals A| ( 12 ,32 ))

but not open in X.Also, ( 12 , 1] is a subspace-neighbourhood of 1

(since it equals A| ( 12 ,32 ))

but is not a neighbourhood of 1 in X.

2.20 Let A be 3-open in X,B ( A) be 3A-open.Then B = A| G for some G g 3the intersection of two 3-open sets,

therefore 3-open.Let A be 3-closed in X,B ( A) be 3A-closed.Then B = A| F for some 3-closed F

the intersection of two 3-closed sets,therefore 3-closed.

2.21 Let (A, 3A) be a subspace ofmetrisable (X, 3).Choose a metric d on X so that 3d is 3.Then d|AA is a metric on A. Call it d.We claim: 3d is 3A.

Let p g A,N A. Then:

N is a 3d-neighbourhood of pK M% > 0 for which {q g A : d(p, q) < %} NK M% > 0 such that A| {q g X : d(p, q) < %} NK N contains the trace onto A of a 3d-neighbourhood of pK N is the trace onto A of a 3d-neighbourhood of pK N is a 3A-neighbourhood of p.

Since 3d and 3A yield the same neighbourhoods of points of A, they are thesame topology (see, for example, 2.6).

2.22 Let (X, 3) satisfy every non-empty open set is uncountable.Let (A, 3A) be one of its subspaces, where A is open.Any 3A-open (non-empty) set is also 3-open (see 2.20),


therefore uncountable,so (A, 3A) possesses the same property.

(R, 3usual) satises the condition every non-empty open set is uncountablebut its subspace Z certainly does not!

Let (X, 3) satisfy every countable set is closed.Let (A, 3A) be one of its subspaces.Any C A that is countableis countable as a subset of Xand is therefore closed in 3.Now C = A| C is also 3A-closed.So (A, 3A) enjoys the same property.

3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Continuity andconvergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Continuity

3.1 Definition A mapping f : (X, 3) C (Y , 3) from a topological space to atopological space is called continuous if

G g 3 I f 1(G) g 3;

that is, if the inverse image of every open set is open.

There are several dierent ways to recognise continuity; here are two of them:

3.2 Lemma For a mapping f : (X, 3)C (Y , 3), the following are equivalent:

(i) f is continuous;(ii) for each 3-closed subset K of Y , f 1(K) is 3-closed in X;(iii) for each A X, f (A) f (A).

3.3 Examples

(i) A map from a metric space to a metric space is continuous (in the metricsense) if and only if it is continuous between the topological spaces inducedby the metrics. Therefore, all the examples of continuity you knew earlier,you still know!

(ii) Any map whose domain is discrete must be continuous. Any map whosecodomain is trivial must be continuous.

(iii) Constant maps are continuous.

3.4 Lemma The composite of two continuous maps is continuous.

3.5 Lemma Any restriction of a continuous map is continuous.

3.6 Lemma Given a mapping f : (X, 3)C (Y , 3) and a set Z satisfying f (X) Z Y , let f be the same map as f except that its codomain is Z. Then f iscontinuous if and only if f is continuous.

30 3 CONTINUITY AND CONVERGENCE

3.7 Examples

(i) Any one-to-one map from (R, 3cc) to (R, 3cf) is continuous.(ii) Any continuous map from (R, 3cf) to (R, 3usual) is constant.

3.8 Definition A mapping f : (X, 3)C (Y , 3) is called a homeomorphism if

(i) it is one-to-one and onto, and(ii) it is continuous, and(iii) the inverse map f 1 is continuous.

Whenever such a map exists between two spaces, they are called homeomorphic,which really means that they have identically the same behaviour as topologicalspaces (compare isomorphism in groups or linear isomorphism in vectorspaces).

A topological property or homeomorphic invariant is a statement about topo-logical spaces which, whenever it is true for a space (X, 3), is necessarily truealso for any space that is homeomorphic with (X, 3). In eect, these are theproperties/statements that can be expressed purely in terms of open sets plus settheory. Two spaces are homeomorphic if and only if they have exactly the sametopological properties.

3.9 Examples

(i) The identity map on a space is a homeomorphism. The composite of twohomeomorphisms is a homeomorphism. The inverse of a homeomorphismis a homeomorphism. Being homeomorphic is therefore an equivalencerelation on any set of spaces.

(ii) Any two open intervals on the real line (R, 3usual) are homeomorphic to oneanother, and are homeomorphic to the entire real line.

(iii) (R, 3usual) and (R, 3cf) are not homeomorphic.

3.10 Illustration of homeomorphism: the fractal character of the Cantor setsNotice that the subset Cantor10 of the real line that we described in 1.10 cannotcontain any interval: for at stage n of the process, the initial interval of length 1has been reduced to n[0, 1], the union of 2n disjoint subintervals each of length10n; therefore if J denotes a non-degenerate interval of positive length $, we needonly choose a positive integer n such that 10n < $ to deduce that J cannot becontained in n[0, 1], let alone in Cantor10 itself.

Consider now a typical one of the intervals created at stage n of the iteration: say,[p, q]. All the stage-n intervals have the same length 10n, so q = p + 10n andthe map x C x p transforms [p, q] onto [0, 10n]. Furthermore, since p as adecimal consists of an initial block of exactly n-many 0s and 9s, followed by anendless stream of 0s (which, as in conventional arithmetic, we can safely ignore),subtraction of p does not change any decimal digit after the nth. It follows that

CONTINUITY 31

x C x p actually transforms Cantor10 | [p, q] onto Cantor10 | [0, 10n]. Byessentially the same argument, x C (10n)x transforms Cantor10 | [0, 10n] ontoCantor10 itself. Since both of these transformations are continuous and continu-ously reversible, we conclude that the part of Cantor10 that falls inside an arbitraryinterval created at some stage in the iterative process that we described is homeo-morphic to the entire space. The critical point to make is that, if x is any point ofCantor10 and N is any Cantor10-neighbourhood of x, then we can select % > 0 sosmall that Cantor10 | (x %, x + %) lies inside N, and now x must belong to aninterval (created at some stage in the process) of length less than %. Thus Cantor10enjoys one of the characteristic properties of the so-called fractals: every neigh-bourhood of a typical point contains a homeomorphic copy of the whole space.That the same is true for the other Cantor sets is best seen by establishing thefollowing result:

3.11 Proposition Cantor excluded-middle sets arising from dierent arithmet-ical bases are homeomorphic.

(Suggested method) There is a natural bijection ( from Cantor3 to Cantor10: it isthe act of replacing 2 by 9 in the tresimal representation of each element ofCantor3and of interpreting the result as a decimal. To be slightly more formal, it is

(( an

3n)=(9an/2

10n

).

Since ( merely substitutes 9s for 2s, it preserves the block structures by whichwe built Cantor3 and Cantor10: for instance, if x falls in the 17th block from theleft created at stage 11 in our iterative construction for Cantor3, then ((x) mustbelong to the 17th block from the left created at stage 11 when we built Cantor10(and vice versa). Notice the lengths (10n for Cantor10, 3n for Cantor3) of thesubinterval blocks that are created at stage n as we construct towards these fractals,and that the gaps between these blocks are in every case at least as big as the blocksthemselves.

Because of Proposition 3.11, it is safe henceforth to refer to the Cantor set when-ever we intend merely to focus on its properties as a topological space (rather thanexplicitly as a subset of the real line) without reference to which arithmetical basewe use in constructing it.

We now briey consider two other properties of mappings, which resemblecontinuity but are less important.

3.12 Definitions

(i) A mapping f : (X, 3)C (Y , 3) is called open if

G g 3 I f (G) g 3,

that is, if the direct image of every open set is open.


(ii) A mapping f : (X, 3)C (Y , 3) is called closed if

K is 3-closedI f (K) is 3-closed,

that is, if the direct image of every closed set is closed.

3.13 Examples(i) Any map into a discrete space is both an open map and a closed map. Any

onto map with trivial domain space is both an open map and a closed map.(ii) Any onto map from (R, 3cf) to (R, 3cc) is a closed map.

3.14 Proposition For a one-to-one and onto mapping f : (X, 3) C (Y , 3), thefollowing are equivalent:

(i) f is continuous,(ii) f 1 is an open map,(iii) f 1 is a closed map.

3.15 Corollary For a one-to-one, onto and continuous mapping f : (X, 3) C(Y , 3), the following are equivalent:

(i) f is a homeomorphism,(ii) f is an open map,(iii) f is a closed map.

Convergent sequences

3.16 Definition A sequence (xn)n1 of points in a topological space (X, 3) is saidto converge to a limit l in X if:

for each neighbourhood U of l, M n(U) g Nsuch that n n(U)I xn g U.

We then write xn C l (Fig. 3.1).

x1 x2 ...

l

xn(U)

U

Fig. 3.1 The sequence (xn)n1 converges to l (see 3.16).

CONVERGENT SEQUENCES 33

3.17 Examples

(i) This incorporates sequential convergence in the senses used in real analysis,complex analysis and metric space theory. Thus, all the examples you knowalready are still valid in the topological setting.

(ii) In a trivial space, every sequence converges to every point as a limit! So theuniqueness of limit that we have come to depend upon in metric spacesdoes not hold good in topological spaces (at least, not in all topologicalspaces).

(iii) In any space, if (xn)n1 is eventually constant (that is, if there is somepositive integer n0 such that for every n n0 we get xn = xn0 ), then (xn)n1certainly converges.

(iv) In a discrete space, this is the only way a sequence can converge.(v) In a cocountable space, this is again the only way a sequence can converge.(vi) Think about what (iv) and (v) tell us: that convergent sequences cannot

spot the dierence between, say, (R, 3cc) and (R, 3disc), even though they areobviously very dierent spaces.

Powerful though they are in analysis and in metric spaces, convergent sequencesalready look to be less eective in coping with the structure of general topologicalspaces. The theme continues below.

3.18 Proposition If G is open in a space (X, 3) then:

whenever x g G and xn C x, we get xn g G for all suciently large n.(1)

3.19 Proposition If, in (X, 3), there is a sequence of elements all belonging to thesubset A and converging to l g X, then l g A.

3.20 Proposition If the mapping f : (X, 3)C (Y , 3) is continuous, then:

whenever xn C l in X, then f (xn)C f (l) in Y . (2)

You probably know that in real/complex analysis and in metric spaces, the con-verses of 3.18, 3.19 and 3.20 are true as well. The main point to make now is that,in general topological spaces, all three converses are false.

3.21 Example For instance, in (R, 3cc), the converses of the last three proposi-tions fail.

Fortunately, it is not in metric spaces alone that sequences fully describe what ishappening topologically. We next identify a broader category of spaces in whichthey are just as eective.


3.22 Definition Let (X, 3) be a space and p an element of X. A countable localbase at p is a sequence

N1,N2,N3, . . .

of neighbourhoods of p such that

(i) N1 N2 N3 . . . and(ii) every neighbourhood of p contains one of the Ni (Fig. 3.2).

N1N2N3p

U

Nn(U)

Fig. 3.2 (N1,N2,N3, . . .) is a countable local base at p (see 3.22).

3.23 Example In any metric space, at any point p, the open balls B(p, 1/n) forn 1 form a countable local base.

3.24 Definition A topological space (X, 3) is called rst-countable if, at each ofits points, there is a countable local base.

For instance, any metrisable space is rst-countable. Now we shall make the pointthat sequences are good for rst-countable spaces, which is the underlying reasonwhy they are good for metric spaces.

3.25 Proposition A subset G of a rst-countable space (X, 3) is open if andonly if:

whenever x g G and xn C x, we get xn g G for all suciently large n.(1)

3.26 Proposition If, in a rst-countable space (X, 3), p is an element and A is asubset, then p g A if and only if there is a sequence of elements all belonging to Athat converges to p.

3.27 Proposition Suppose that there is given a mapping f : (X, 3) C (Y , 3),where (Y , 3) is rst-countable. Then f is continuous if and only if:

whenever xn C l in X, then f (xn)C f (l) in Y . (2)

NETS 35

Nets

There is a way to get round the failure of sequences fully to describe (non-rst-countable) topological spaces, and it amounts to changing the denition ofsequence so as to allow the underlying ordered set which for sequences is alwaysthe positive integers, naturally ordered to be bigger and/or more complicated.

3.28 Definition A directed set (D,) is a non-empty set D together with a qua-siorder in which each two elements possess a common upper bound. That is,is reexive and transitive, and for every a, b g D there is u g D such that botha u and b u.

3.29 Examples The positive integers with their natural (total) order form a dir-ected set. Any ordinal is a directed set. The positive integers under divisibility forma directed set (for instance, the LCM of a and b will serve here as an upper boundfor a and b). Most importantly, if X is any given topological space, x is any givenpoint of X and N(x) is the set of all neighbourhoods of x in X, and we choose toorder N(x) by inverse inclusion that is, N1 N2 means N2 N1 then N(x)becomes a directed set: for any two neighbourhoods of x, their intersection will bea common upper bound.

3.30 Definition A net in a topological space X is a map from some directed setD into X. We normally denote a typical net not by a function-style notation suchas x : D C X but by a notation such as (x#)#gD, partly to bring out the (many)parallels between this notion and that of a sequence in X.

3.31 Definition A net (x#)#gD of points in a topological space (X, 3) is said toconverge to a limit l in X if:

for each neighbourhood U of l, M#(U) g Dsuch that # #(U)I x# g U.

We then write x# C l.

(We often use the informal phrase for all suciently large # to mean for all #that are some threshold value #(U) in the underlying directed set. Thus thecondition in 3.31 can be verbalised as each neighbourhood of l contains x# for allsuciently large values of #.)

It should be obvious that net convergence incorporates sequence convergenceas we described it earlier, but it goes much further by allowing us to adjust theunderlying set of labels no longer constrained to be merely N all the time! so as to cope with points whose neighbourhood structures are complicated. Thefollowing makes that point more formally:

3.32 Lemma Given a point x in a topological space (X, 3), and using the symbolN(x) to stand for the family of all neighbourhoods of x in X, suppose that we


arbitrarily select from eachN inN(x) an element xNgN. Then the net (xN)NgN(x)converges to x.

It is absolutely routine to check that 3.18, 3.19 and 3.20 remain true if we replacesequence by net throughout. Much more interestingly, their converses becometrue when we use nets in place of sequences; that is to say:

3.33 Proposition If G is a subset of a space (X, 3) such that

whenever x g G and a net x# C x, we get x# g G for all suciently large #,(1)

then G is open.

3.34 Proposition If, in a space (X, 3), p g A, then there is a net of elements allbelonging to the subset A that converges to p.

3.35 Proposition If a mapping f : (X, 3)C (Y , 3) satises the condition

whenever a net x# C l in X, then f (x#)C f (l) in Y , (2)

then f is continuous.

Therefore the nice thing about these converses also being valid is that nets fullydescribe what goes on in topological spaces just as sequences do in metric spacesand in rst-countable spaces; that is:

3.36 Proposition A subset G of a topological space (X, 3) is open if and only if:

whenever x g G and a net x# C x, we get x# g G for all suciently large #.(1)

3.37 Proposition If, in a space (X, 3), p is an element and A is a subset, thenp g A if and only if there is a net of elements all belonging toA that converges to p.

3.38 Proposition Consider a mapping f : (X, 3) C (Y , 3). Then f is continu-ous if and only if:

whenever x# C l in X, then f (x#)C f (l) in Y . (2)

Filters

By this stage, we hope to have persuaded the reader that nets provide a natural andfairly intuitive way of importing the notion of convergence to a limit into generaltopological spaces in a manner that is as eective in describing them as sequen-tial convergence is in describing metric spaces. However, it is not the only way.A widely used alternative is the idea of a lter. Nets and lters do very much the

FILTERS 37

same job in general topology, and it seems best to concentrate on only one ofthem when studying the discipline for the rst time. However, the student whointends to go further in topology or its applications will encounter lters sooneror later, and so a very brief introduction to them at this point may ultimately paydividends.

3.39 Definition A non-empty family F of subsets of a non-empty set X is calleda lter on X if it satises the following three conditions:

(i) /g F,(ii) A,B g FI A| B g F,(iii) (A g F,A S X)I S g F.

3.40 Examples

(i) The collection of all supersets of any chosen non-empty A X is a lter onX. The family N(x) of neighbourhoods of a chosen point x in a topologicalspace (X, 3) is a lter on X.

(ii) The complements of the bounded subsets of the coordinate plane (or,indeed, of any given unbounded metric space) comprise a lter on it. Thesubsets D of R for which there exists a g R such that D (, a]comprise a lter on R.

(iii) The collection of conite subsets of an innite set X (that is, 3cfo{}) is alter on X. Likewise, 3cco{} is a lter on any uncountable set.

Filters on the same set may be compared by simple set inclusion and, in particular,a lter on a topological space may be compared with the lter of neighbourhoodsof a point:

3.41 Definition

(i) If F,F are lters on a set X such that F F, we call F a renement of F,and say that F is ner than F.

(ii) A lter F on a topological space (X, 3) is said to converge to a limit p g X ifF N(p); equivalently, if every neighbourhood of p contains a member ofthe lter F.

Here are two indicators of how lter convergence can be used to describe thebehaviour of a topological space:

3.42 Proposition An element p of a topological space (X, 3) lies in the closureof a subset A of X if and only if there is a lter F on X such that A g F and Fconverges to p.

3.43 Proposition A topological space (X, 3) is compact if and only if every lteron X can be rened to a convergent lter (on X).


3.44 Note Some insight into the relationship between lters and nets on atopological space (X, 3) is oered here.

(i) Given a net x = (x#)#gD (where D is, as usual, a directed set) and #0 g D, bythe #0th tail of the net we mean the set {x# : # #0}. Dene F(x) to consistof all subsets of X that contain a tail of the net x. Then F(x) is a lter on X.Furthermore, F(x) converges to a limit p if and only if the net x convergesto p.

(ii) Given a lter F on X, notice that F is already a directed set under inverse setinclusion. For each A g F, choose an element xA arbitrarily in A. Then(xA)AgF is a net in X. Furthermore, F converges to a limit p if and only ifevery such net (xA)AgF converges to p.

Corresponding approximately to the way in which we used countable local baseto extract the essential structure of a system of neighbourhoods, it is often con-venient to select out enough elements from a lter to be able to describe theentire thing:

3.45 Definitions

(a) A non-empty familyP of subsets of a non-empty set X is called a lterbase onX if it satises the following two conditions:(i) /g P,(ii) A,B g PI MC g P such that C A| B.

(b) When P is a lterbase on X, the family {G X : M F g P such that F G}is a lter on X. Then P is said to be a base for this lter.

(c) A lterbase (on a topological space) is said to converge to a point p if the lterfor which it is a base converges to p (equivalently, if every neighbourhood ofp contains a member of the lterbase).

One of the (several) reasons why lterbases are used as well as or instead of ltersis that the image of a lter under a mapping is quite often not a lter that is, iff : X C Y and F is a lter on X, then f (F) = {f (A) : A g F} may fail to be alter on Y whereas this issue does not arise with lterbases: if F here is a lter-base, then f (F) is also a lterbase. This facilitates the following characterisation ofcontinuous maps via lter convergence:

3.46 Proposition Let f : (X, 3) C (Y , 2) be a mapping between topologicalspaces. Then f is continuous if and only if, whenever P is a lterbase on Xconverging to a limit p g X, then the lterbase f (P) converges to f (p) in (Y , 2).

Exercises Essential Exercises 14, 16, 1829 and 47 are based on the material inthis chapter. It is particularly recommended that you should try numbers 19, 20,21, 24, 26, 27 and 29.



3.2 (i)I (ii)Suppose f continuous.If K Y is 3-closedthen Y \K is 3-open.Therefore f 1(Y \K) is 3-open,that is, X \ f 1(K) is 3-open,therefore f 1(K) is 3-closed.

(ii)I (iii)Suppose f 1(each 3-closed set) is 3-closed.Given A X : f (A) is 3-closed,therefore f 1(f (A)) is 3-closed.But A f 1(f (A)) f 1(f (A),therefore A f 1(f (A)),that is, f (A) f (A).

(iii)I (i)Suppose f (A) f (A) for all A X.Given open H Y , try A = X \ f 1(H) = f 1(Y \H). So:

f (f 1(Y \H)) f (f 1(Y \H)) Y \H = Y \H,

that is, f 1(Y \H) f 1(Y \H),therefore f 1(Y \H) is already 3-closed,

that is, X \ f 1(H) is 3-closed,that is, f 1(H) is 3-open.

So f is continuous.

3.4 Let f : (X, 3) (Y , 3), g : (Y , 3) (Z, 3) both be continuous.For any 3-open J Z, g1(J) is 3-open.Therefore f 1(g1(J)) is 3-open.That is, (g f )1(J) is 3-open.Therefore g f : X Z is continuous.

3.5 Let:

f : (X, 3) (Y , 3) be continuous, = A X,g = f |A be the restriction of f to A.


For any 3-open H Y ,

g1(H) = {a g A : f (a) = g(a) g H}= A| f 1(H),which is 3A-open.So g is continuous.

3.6 The 3Z-open sets take the form Z |H, where H g 3.

Now (f )1(Z | H) and f 1(H) are the same set!So continuity of f and continuity o

0198702337 Topology c

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