Table Of ContentTopology
James Munkres
Second Edition
ISBN 10: 1-292-02362-7
ISBN 13: 978-1-292-02362-5
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ISBN 10: 1-292-02362-7
ISBN 13: 978-1-292-02362-5
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P E A R S O N C U S T O M L I B R AR Y
Table of Contents
Chapter 1. Set Theory and Logic
James Munkres 1
Chapter 2. Topological Spaces and Continuous Functions
James Munkres 73
Chapter 3. Connectedness and Compactness
James Munkres 145
Chapter 4. Countability and Separation Axioms
James Munkres 187
Chapter 5. The Tychonoff Theorem
James Munkres 228
Chapter 6. Metrization Theorems and Paracompactness
James Munkres 241
Chapter 7. Complete Metric Spaces and Function Spaces
James Munkres 261
Chapter 8. Baire Spaces and Dimension Theory
James Munkres 292
Chapter 9. The Fundamental Group
James Munkres 317
Chapter 10. Separation Theorems in the Plane
James Munkres 372
Chapter 11. The Seifert-van Kampen Theorem
James Munkres 403
Chapter 13. Classification of Covering Spaces
James Munkres 443
Chapter 12. Classification of Surfaces
James Munkres 468
Bibliography
James Munkres 499
I
Chapter 1
Set Theory and Logic
We adopt, as most mathematicians do, the naive point of view regarding set theory.
Weshallassumethatwhatismeantbyasetofobjectsisintuitivelyclear,andweshall
proceedonthatbasiswithoutanalyzingtheconceptfurther. Suchananalysisproperly
belongstothefoundationsofmathematicsandtomathematicallogic,anditisnotour
purposetoinitiatethestudyofthosefields.
Logicians have analyzed set theory in great detail, and they have formulated ax-
ioms for the subject. Each of their axioms expresses a property of sets that mathe-
maticians commonly accept, and collectively the axioms provide a foundation broad
enoughandstrongenoughthattherestofmathematicscanbebuiltonthem.
It is unfortunately true that careless use of set theory, relying on intuition alone,
can lead to contradictions. Indeed, one of the reasons for the axiomatization of set
theory was to formulate rules for dealing with sets that would avoid these contradic-
tions. Although we shall not deal with the axioms explicitly, the rules we follow in
dealingwithsetsderivefromthem. Inthisbook,youwilllearnhowtodealwithsets
in an “apprentice” fashion, by observing how we handle them and by working with
themyourself. Atsomepointofyourstudies,youmaywishtostudysettheorymore
carefullyandingreaterdetail;thenacourseinlogicorfoundationswillbeinorder.
From Chapter 1 of Topology, Second Edition. James R. Munkres.
Copyright © 2000 by Pearson Education, Inc. All rights reserved.
1
4 SetTheoryandLogic Ch.1
§1 Fundamental Concepts
Here we introduce the ideas of set theory, and establish the basic terminology and
notation. Wealsodiscusssomepointsofelementarylogicthat,inourexperience,are
apttocauseconfusion.
BasicNotation
Commonlyweshallusecapitalletters A, B, ... todenotesets, andlowercaseletters
a, b, ... to denote the objects or elements belonging to these sets. If an object a
belongstoaset A,weexpressthisfactbythenotation
a ∈ A.
Ifa doesnotbelongto A,weexpressthisfactbywriting
a ∈/ A.
Theequalitysymbol=isusedthroughoutthisbooktomeanlogicalidentity. Thus,
whenwewritea =b,wemeanthat“a”and“b”aresymbolsforthesameobject. This
iswhatonemeansinarithmetic,forexample,whenonewrites 2 = 1. Similarly,the
4 2
equation A = B statesthat“A”and“B”aresymbolsforthesameset;thatis, Aand B
consistofpreciselythesameobjects.
Ifa andbaredifferentobjects,wewritea (cid:4)= b;andif Aand B aredifferentsets,
wewrite A (cid:4)= B. Forexample,if A isthesetofallnonnegativerealnumbers,and B
isthesetofallpositiverealnumbers,then A (cid:4)= B,becausethenumber0belongsto A
andnotto B.
Wesaythat Aisasubsetof B ifeveryelementof Aisalsoanelementof B;and
weexpressthisfactbywriting
A ⊂ B.
Nothinginthisdefinitionrequires Atobedifferentfrom B;infact,if A = B,itistrue
thatboth A ⊂ B and B ⊂ A. If A ⊂ B and A isdifferentfrom B,wesaythat A isa
propersubsetof B,andwewrite
A (cid:2) B.
The relations ⊂ and (cid:2) are called inclusion and proper inclusion, respectively. If
A ⊂ B,wealsowrite B ⊃ A,whichisread“B contains A.”
How does one go about specifying a set? If the set has only a few elements, one
cansimplylisttheobjectsintheset,writing“Aisthesetconsistingoftheelementsa,
b,andc.” Insymbols,thisstatementbecomes
A ={a,b,c},
wherebracesareusedtoenclosethelistofelements.
2
§1 FundamentalConcepts 5
Theusualwaytospecifyaset,however,istotakesomeset Aofobjectsandsome
property that elements of A may or may not possess, and to form the set consisting
of all elements of A having that property. For instance, one might take the set of
real numbers and form the subset B consisting of all even integers. In symbols, this
statementbecomes
B ={x | x isaneveninteger}.
Here the braces stand for the words “the set of,” and the vertical bar stands for the
words “such that.” The equation is read “B is the set of all x such that x is an even
integer.”
TheUnionofSetsandtheMeaningof“or”
Giventwosets Aand B,onecanformasetfromthemthatconsistsofalltheelements
of A togetherwithalltheelementsof B. Thissetiscalledtheunionof A and B and
isdenotedby A∪ B. Formally,wedefine
A∪ B ={x | x ∈ Aorx ∈ B}.
Butwemustpauseatthispointandmakesureexactlywhatwemeanbythestatement
“x ∈ Aorx ∈ B.”
In ordinary everyday English, the word “or” is ambiguous. Sometimes the state-
ment “P or Q” means “P or Q, or both” and sometimes it means “P or Q, but not
both.” Usuallyonedecidesfromthecontextwhichmeaningisintended. Forexample,
supposeIspoketotwostudentsasfollows:
“MissSmith,everystudentregisteredforthiscoursehastakeneitheracoursein
linearalgebraoracourseinanalysis.”
“Mr.Jones,eitheryougetagradeofatleast70onthefinalexamoryouwillflunk
thiscourse.”
Inthecontext,MissSmithknowsperfectlywellthatImean“everyonehashadlinear
algebra or analysis, or both,” and Mr. Jones knows I mean “either he gets at least 70
or he flunks, but not both.” Indeed, Mr. Jones would be exceedingly unhappy if both
statementsturnedouttobetrue!
In mathematics, one cannot tolerate such ambiguity. One has to pick just one
meaningandstickwithit,orconfusionwillreign. Accordingly,mathematicianshave
agreedthattheywillusetheword“or”inthefirstsense,sothatthestatement“P orQ”
alwaysmeans“P or Q,orboth.” Ifonemeans“P or Q,butnotboth,”thenonehasto
includethephrase“butnotboth”explicitly.
Withthisunderstanding,theequationdefining A∪Bisunambiguous;itstatesthat
A∪ B isthesetconsistingofallelementsx thatbelongto Aorto B ortoboth.
3
6 SetTheoryandLogic Ch.1
TheIntersectionofSets,theEmptySet,andtheMeaningof“If... Then”
Givensets A and B,anotherwayonecanformasetistotakethecommonpartof A
andB. Thissetiscalledtheintersectionof AandBandisdenotedby A∩B. Formally,
wedefine
A∩ B ={x | x ∈ Aandx ∈ B}.
Butjustaswiththedefinitionof A∪B,thereisadifficulty. Thedifficultyisnotinthe
meaning of the word “and”; it is of a different sort. It arises when the sets A and B
happentohavenoelementsincommon. Whatmeaningdoesthesymbol A∩ B have
insuchacase?
To take care of this eventuality, we make a special convention. We introduce a
special set that we call the empty set, denoted by ∅, which we think of as “the set
havingnoelements.”
Usingthisconvention,weexpressthestatementthat Aand B havenoelementsin
commonbytheequation
A∩ B =∅.
Wealsoexpressthisfactbysayingthat Aand B aredisjoint.
Nowsomestudentsarebotheredbythenotionofan“emptyset.” “How,”theysay,
“can you have a set with nothing in it?” The problem is similar to that which arose
manyyearsagowhenthenumber0wasfirstintroduced.
The empty set is only a convention, and mathematics could very well get along
without it. But it is a very convenient convention, for it saves us a good deal of
awkwardness in stating theorems and in proving them. Without this convention, for
instance, one would have to prove that the two sets A and B do have elements in
commonbeforeonecouldusethenotation A∩ B. Similarly,thenotation
C ={x | x ∈ Aandx hasacertainproperty}
could not be used if it happened that no element x of A had the given property. It is
muchmoreconvenienttoagreethat A∩ B andC equaltheemptysetinsuchcases.
Sincetheemptyset∅ismerelyaconvention,wemustmakeconventionsrelating
it to the concepts already introduced. Because ∅ is thought of as “the set with no
elements,”itisclearweshouldmaketheconventionthatforeachobjectx,therelation
x ∈∅doesnothold. Similarly,thedefinitionsofunionandintersectionshowthatfor
everyset Aweshouldhavetheequations
A∪∅= A and A∩∅=∅.
The inclusion relation is a bit more tricky. Given a set A, should we agree that
∅⊂ A? Oncemore,wemustbecarefulaboutthewaymathematiciansusetheEnglish
language. Theexpression∅ ⊂ A isashorthandwayofwritingthesentence, “Every
element that belongs to the empty set also belongs to the set A.” Or to put it more
4
§1 FundamentalConcepts 7
formally,“Foreveryobjectx,ifx belongstotheemptyset,thenx alsobelongstothe
set A.”
Is this statement true or not? Some might say “yes” and others say “no.” You
willneversettlethequestionbyargument, onlybyagreement. Thisisastatementof
the form “If P, then Q,” and in everyday English the meaning of the “if ... then”
construction is ambiguous. It always means that if P is true, then Q is true also.
Sometimesthatisallitmeans;othertimesitmeanssomethingmore: thatif P isfalse,
Q mustbefalse. Usuallyonedecidesfromthecontextwhichinterpretationiscorrect.
Thesituationissimilartotheambiguityintheuseoftheword“or.” Onecanrefor-
mulate the examples involving Miss Smith and Mr. Jones to illustrate the ambiguity.
SupposeIsaidthefollowing:
“Miss Smith, if any student registered for this course has not taken a course in
linearalgebra,thenhehastakenacourseinanalysis.”
“Mr.Jones, ifyougetagradebelow70onthefinal, youaregoingtoflunkthis
course.”
Inthecontext,MissSmithunderstandsthatifastudentinthecoursehasnothadlinear
algebra,thenhehastakenanalysis,butifhehashadlinearalgebra,hemayormaynot
havetakenanalysisaswell. AndMr.Jonesknowsthatifhegetsagradebelow70,he
willflunkthecourse,butifhegetsagradeofatleast70,hewillpass.
Again, mathematics cannot tolerate ambiguity, so a choice of meanings must be
made. Mathematicians have agreed always to use “if ... then” in the first sense, so
thatastatementoftheform“If P,then Q”meansthatif P istrue, Q istruealso,but
if P isfalse, Q maybeeithertrueorfalse.
Asanexample,considerthefollowingstatementaboutrealnumbers:
Ifx >0,thenx3 (cid:4)=0.
It is a statement of the form, “If P, then Q,” where P is the phrase “x > 0” (called
thehypothesisofthestatement)and Q isthephrase“x3 (cid:4)= 0”(calledtheconclusion
ofthestatement). Thisisatruestatement, forineverycaseforwhichthehypothesis
x >0holds,theconclusionx3 (cid:4)=0holdsaswell.
Anothertruestatementaboutrealnumbersisthefollowing:
Ifx2 <0,thenx =23;
ineverycaseforwhichthehypothesisholds,theconclusionholdsaswell. Ofcourse,
it happens in this example that there are no cases for which the hypothesis holds. A
statementofthissortissometimessaidtobevacuouslytrue.
To return now to the empty set and inclusion, we see that the inclusion ∅ ⊂ A
does hold for every set A. Writing ∅ ⊂ A is the same as saying, “If x ∈ ∅, then
x ∈ A,”andthisstatementisvacuouslytrue.
5
8 SetTheoryandLogic Ch.1
ContrapositiveandConverse
Our discussion of the “if ... then” construction leads us to consider another point of
elementary logic that sometimes causes difficulty. It concerns the relation between a
statement,itscontrapositive,anditsconverse.
Given a statement of the form “If P, then Q,” its contrapositive is defined to be
thestatement“If Q isnottrue,then P isnottrue.” Forexample,thecontrapositiveof
thestatement
Ifx >0,thenx3 (cid:4)=0,
isthestatement
Ifx3 =0,thenitisnottruethatx >0.
Notethatboththestatementanditscontrapositivearetrue. Similarly,thestatement
Ifx2 <0,thenx =23,
hasasitscontrapositivethestatement
Ifx (cid:4)=23,thenitisnottruethatx2 <0.
Again,botharetruestatementsaboutrealnumbers.
Theseexamplesmaymakeyoususpectthatthereissomerelationbetweenastate-
mentanditscontrapositive. Andindeedthereis;theyaretwowaysofsayingprecisely
thesamething. Eachistrueifandonlyiftheotheristrue; theyarelogicallyequiva-
lent.
This fact is not hard to demonstrate. Let us introduce some notation first. As a
shorthandforthestatement“If P,then Q,”wewrite
P (cid:9)⇒ Q,
whichisread“P implies Q.” Thecontrapositivecanthenbeexpressedintheform
(not Q)(cid:9)⇒(not P),
where“not Q”standsforthephrase“Q isnottrue.”
Nowtheonlywayinwhichthestatement“P ⇒ Q”canfailtobecorrectisifthe
hypothesis P istrueandtheconclusion Q isfalse. Otherwiseitiscorrect. Similarly,
the only way in which the statement (not Q) ⇒ (not P) can fail to be correct is if
the hypothesis “not Q” is true and the conclusion “not P” is false. This is the same
as saying that Q is false and P is true. And this, in turn, is precisely the situation in
which P ⇒ Q failstobecorrect. Thus,weseethatthetwostatementsareeitherboth
correct or both incorrect; they are logically equivalent. Therefore, we shall accept a
proofofthestatement“not Q ⇒not P”asaproofofthestatement“P ⇒ Q.”
There is another statement that can be formed from the statement P ⇒ Q. It is
thestatement
Q (cid:9)⇒ P,
6
