Table Of ContentUndergraduate Texts in Mathematics
John B. Conway
A Course in
Point Set
Topology
Undergraduate Texts in Mathematics
Undergraduate Texts in Mathematics
SeriesEditors:
SheldonAxler
SanFranciscoStateUniversity,SanFrancisco,CA,USA
KennethRibet
UniversityofCalifornia,Berkeley,CA,USA
AdvisoryBoard:
ColinAdams,WilliamsCollege,Williamstown,MA,USA
AlejandroAdem,UniversityofBritishColumbia,Vancouver,BC,Canada
RuthCharney,BrandeisUniversity,Waltham,MA,USA
IreneM.Gamba,TheUniversityofTexasatAustin,Austin,TX,USA
RogerE.Howe,YaleUniversity,NewHaven,CT,USA
DavidJerison,MassachusettsInstituteofTechnology,Cambridge,MA,USA
JeffreyC.Lagarias,UniversityofMichigan,AnnArbor,MI,USA
JillPipher,BrownUniversity,Providence,RI,USA
FadilSantosa,UniversityofMinnesota,Minneapolis,MN,USA
AmieWilkinson,UniversityofChicago,Chicago,IL,USA
Undergraduate Texts in Mathematics are generally aimed at third- and fourth-
year undergraduate mathematics students at North American universities. These
texts strive to provide students and teachers with new perspectives and novel
approaches. The books include motivation that guides the reader to an apprecia-
tionofinterrelationsamongdifferentaspectsofthesubject. Theyfeatureexamples
thatillustratekeyconceptsaswellasexercisesthatstrengthenunderstanding.
Forfurthervolumes:
http://www.springer.com/series/666
John B. Conway
A Course in Point
Set Topology
123
JohnB.Conway
DepartmentofMathematics
TheGeorgeWashingtonUniversity
Washington,DC,USA
ISSN0172-6056 ISSN2197-5604(electronic)
ISBN978-3-319-02367-0 ISBN978-3-319-02368-7(eBook)
DOI10.1007/978-3-319-02368-7
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013951651
MathematicsSubjectClassification(2010):54A05,54E35,54D20,58C07
©SpringerInternationalPublishingSwitzerland2014
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For Ann
How could I be so lucky!
Preface
Pointset topology was my first love in mathematics. I took the courseas an
undergraduateatLoyolaUniversityinNew Orleansandmyprofessor,Harry
Fledderman, told me to go to the library and solve all the problems in the
book while he tutored the other student who had signed up for the course.
(Yes,Iknowitsoundsstrangetoday,buttherewereonlytwostudentsinthe
course.) I kept a notebook with my solutions, and once a week I reported
for his inspection of my work. I felt like a real mathematician learning real
mathematics. Ithad a greatinfluence onme andmade me realizehow much
Iwantedtobeamathematician. EvennowIcan’ttellyouwhethertheloveI
havefor point set topologywas the cause of this feeling or whether that love
wasa consequenceof this learningstyle. I was disappointedto later discover
that researchin this area had mostly petered out. I found equally attractive
research areas in which to sow my oats, but I always retained this youthful
love affair.
You can probably guess that I have long wanted to write a book on
this topic, but other things took precedence. I am glad that was the case
because now I think I have a better approach. I had an epiphany about
halfwaythroughmycareerwhenIrealizedIdidn’thavetoteachmystudents
everythingIhadlearnedaboutthesubjectathand. Ilearnedmathematicsin
schoolthatIneverusedagain,andnotjustbecausethosethingswereinareas
in which I never did research. At least part of this, I suspect, was because
some of my teachers hadn’t had this insight. Another reason is that many
authors write textbooks as though they are writing a monograph directed
at other faculty rather than thinking of the students as the audience. Also,
mathematics refines and refreshes itself with time. Certain topics that were
important at the inception of an area fade in significance, and some that are
useful in various areas today must be added. Other topics are important,
but only if you are one of that small percentage who specialize in a specific
part of research; such things should not be taught to everyone who takes an
vii
viii Preface
introductory course. In addition, when a subject is developing, there is an
emphasis on finding the intellectual boundaries of the concepts. Unless that
viewpoint is abandoned when the subject is taught, it results in a greater
prominence of pathology. An examination of early texts in any subject will
reveal such an emphasis. With time, however, it is crucial to decide what
should be taught in an introductory course such as the kind this book is
written for. I see the purpose of a course in point set topology as giving
the student a set of tools. The material is used in almost every part of
mathematics.
In addition, I have come to believe that in teaching, it is best to go
fromthe particulartothe moregeneral,anapproachthathasdominatedmy
presentations in the last several texts I have written. To begin with, that is
the way mathematics evolves. In addition, my experience is that all but the
very best students find such an approach more digestible. The present book
reflects my belief in this approach.
I see this text as aimed at an undergraduate audience that has had cal-
culus and been exposed to the ideas of basic set theory like subsets, unions,
intersections,functions,andlittleelse. Nevertheless,Ithinkitadvisablethat
they have had at least a semester of analysis and been properly exposed to
convergence and other topological notions in the real line. (I have included
some appendices to help bridge the gap, but I am sure this will not suffice
forall.) Ialsothink pointsettopologyis anexcellentplacetobeginlearning
how to digest and write proofs. Thus, I tend to go slowly at the start of the
book, including more detail than is needed for a seasoned student and even
more than I include later in the text.
Following my philosophy of beginning with the particular, I start with
metric spaces. I believe that these are far easier to connect with students’
experience. Theyalsoseemtometobethemoreprevalenttopologicalspaces
usedinotherareasandarethereforeworthextraemphasis. Chapter2defines
and develops abstract topological spaces, with metric spaces as the source
of inspiration. I narrow the discussion by quickly restricting the focus to
Hausdorff spaces. Needless to say, some of the more elementary arguments
in topological spaces are the same as those in metric spaces. There is no
problemhere; Ijust referstudents to the metric space proofandinvite them
tocarryouttheanalogousargument,whichinmostcasesisalmostidentical.
Chapter 3 concentrates on continuous real-valued functions. My belief
is that the continuous functions on a space are more important than the
underlying space. Maybe that’s because I’m an analyst. I know that much
of moderntopology concentratesonthe underlying geometryof a space,but
surely that must be saved until after the student has encountered the need.
Biographies. In this book Icontinue the practice,startedin a previous
work, of including short biographicalnotes when a mathematician’s result is
mentioned. There is no scholarship on my part in this as all the material
Preface ix
is from secondary sources, principally what I could find on the Web. In
particular, I made heavy use of the site at the University of St Andrews
http://www-history.mcs.st-andrews.ac.uk/history/BiogIndex.html
and Wikipedia. I continue my practice of emphasizing personal aspects of
these lives over their mathematical achievements, especially if there is some-
thing there that interests me.
Style. The longerI ama mathematicianandthe moreI write, the more
I ask myself questions about style. There is a difference between what you
write and how you speak. That’s true in mathematics just as it is outside
ofmathematics. I think I write more informally thanmany mathematicians,
but there are some rules I try to follow even though they are not universal.
OnesuchruleisthatwithessentiallynoexceptionsIusesymbolslike∈,⊆,>,
and so on as verbs and only as verbs. (For example, ∈ translates as “is an
element of” and not as “in.”) I think this consistency expedites reading. I
experimented a long time ago with using such symbols only as prepositions
butquicklydecidedthiswasawkward. WhenIamhavingadiscussioninmy
office, I will frequently use them both ways, but when I write I try to stick
with using them as verbs. So don’t forget to read them that way.
Long ago I realized that every result is not a theorem. The label “The-
orem” is reservedfor the truly important results. It’s not that those labeled
“Proposition” are unimportant, but they may be more routine or, perhaps,
they just don’t have the impact on the development of the subject at hand.
A corollary is a direct consequence of a proposition or a theorem. A lemma
is a result whose usefulness is usually limited to the proof of the next result.
A Word to Students. If you want to learn mathematics, you cannot
approachit as a spectator sport; sitting on the edge of the pool and dipping
yourtoeinwillnotgetyouintothesubject. Youmustjumpintoitwithboth
feet,commityourself,anddoalotofdirtyworkandsplashingaroundbefore
you can enter the profession at any level. In the course of this book there
aremany places where I leaveproofs to readersas exercises;do them. When
I give such an exercise, I think it is well within the scope of your ability
provided you understood the concepts. Doing those exercises will confirm
that you understand what has come before; if you cannot do them, it may
mean you overlookedsomething and should go back.
Throughout the text you will see words like Verify! and Why? I am
trying to put a speed bump in your reading. I want you to be sure you
understand what was just said.
Sometimes my exercises ask a question. A basic part of mathematics is
deciding whether something is true andthen provingit. Mathematicians are
constantly trying to discover whether something is true and are seldom, if
ever, presented with a known truth and asked to prove it. So when you see
such an exercise and you think something is true, you must prove it; if you
think it false, you must find a counterexample. (The ability to manufacture
examples is a precious talent that you should cultivate.)
