Point-Set Topology with Topics
Basic General Topology for Graduate Studies
Robert André
University of Waterloo, Canada
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Library of Congress Cataloging-in-Publication Data
Names: André, Robert (Mathematician), author.
Title: Point-set topology with topics : basic general topology for graduate studies / Robert André, University of Waterloo, Canada.
Description: New Jersey : World Scientific, [2024] | Includes bibliographical references and index.
Identifiers: LCCN 2023021913 | ISBN 9789811277337 (hardcover) | ISBN 9789811277344 (ebook for institutions) | ISBN 9789811277351 (ebook for individuals)
Subjects: LCSH: Topology--Textbooks. | Point set theory--Textbooks.
Classification: LCC QA611 .A537 2024 | DDC 514/.322--dc23/eng20230819
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Preface
Thistextwaspreparedtoserveasanintroductiontothestudyof generaltopology.Moststudentsinmathematicsarerequired,atsome pointintheirstudy,tohaveknowledgeofthefundamentalsofgeneraltopologyandmastertopologicaltechniquesthatmaybeuseful intheirareaofspecialization.“Trustus,notonlywillyoubeglad tobeskilledatusingthesetools,butalotofconceptsyoustudyin yourfieldofinterestwilleventuallybeseenasspecificcasesinvestigatedinamoregeneralcontextingeneraltopology”theyaretold. Itsometimesoccursthatsomestudentsdevelopaparticularfascinationforgeneraltopologyinspiteofitoccasionallybeingdescribed asbeing“nolongerinfashionanymore”.Recallthatmathematicians suchasGeorgeCantorandFelixHausdorff,werealsotoldthatsome ofthemathematicstheyspenttimeinvestigatingwas“notinfashion”.Reasonsforthecontinuedstudyofthistopicoftengobeyond thesimpleperceptionthatitisapracticalorusefultool.Theintricatebeautyofthemathematicalstructuresthatarederivedinthis fieldbecomeitsmainattraction.Thisis,ofcourse,howthiswriter perceivesthesubjectandservedasthemainmotivationtoprepare atextbookthatwillhelpthereaderenjoyitsstudy.Ofcourse,one wouldnaturallyhopethatanauthorwouldwriteabookonlyabout somethingheorshefeelspassionateabout.Ioftenheardsomestudentsdescribegeneraltopologyasbeing“hard”.Well,someofitis. ButIhaveoftenthoughtthatmaybethisperceptionwasdeveloped becausetheydidnotapproachitquiteintherightway.Inthistext, wetrytoimproveonthewaysthatstudentsareintroducedtoit.
Butfirst,Ishouldatleastwriteafewwordsaboutthemathematicalcontentofthistextbook.Thechoiceofcontentaswellas theorderandpaceofthepresentationoftheconceptsfoundinthe textweredevelopedwithseniormathundergraduateormathgraduatestudentsinmind.Thetargetedreaderwillhavebeenexposedto somemathematicalrigortoalevelnormallyfoundinanintroduction tomathematicalanalysistextsoraspresentedinanintroductionto linearalgebraorabstractalgebratexts.ThefirsttwosectionsofPart Iconsistmostlyofareviewintheformofasummarizedpresentation ofverybasicideasonnormedvectorspacesandmetricspaces.These aremeanttoeasethereaderintothemainsubjectmatterofgeneral topology(inChapters3–20ofPartsII–VI).PartsII–VInormally formthecorematerialcontainedinmost,oneortwosemester,Basic GeneralTopologycourse.Oncewehaveworkedthroughthemost fundamentalconceptsoftopologyinChapters 1–20,thereaderwill beexposedtobriefintroductionstomorespecializedoradvanced topics.ThesearepresentedinPartVIIintheformofasequenceof chaptersmanyofwhichcanbereadorstudied,independently,orin shortsequencesoftwoorthreechapters,providedthestudenthas masteredChapters3–20.
Chaptersrelatedtothemorebasicideasofgeneraltopologyare followedbyalistof Conceptreview typequestions.Thesequestions highlightforstudentsthemainideaspresentedinthatsectionand willhelptesttheirunderstandingoftheseconcepts.Theanswers toall Conceptreview questionsareinthemainbodyofthetext. Attemptingtoanswerthesequestionswillhelpthestudentdiscover essentialnotionswhichareoftenoverlookedwhenfirstexposedto theseideas.Readingasectionprovidesacertainlevelofunderstanding,butansweringquestions,evensimpleones,relatedtoits contentrequiresamuchdeeperunderstanding.Theeffortsrequired inansweringcorrectlysuchquestionsleadsthestudenttotheability tosolvemorecomplexproblemsintheExercisesections.Ifthestudentdesiresamorein-depthstudyofatopicinPartVII,thereare manyexcellenttopologybooksthatcansatisfythisneed.
Textbookexampleswillserveassolutionmodelstomostofthe exercisequestionsattheendofeachsection.
Incertainsections,wemakeuseofelementarysettheory.Astudentwhofeelsabitrustywhenfacingtheoccasionalreferencestoset theorynotionsmaywanttoreviewsomeofthese.Forconvenience,
asummaryofthemainsettheoryconceptsappearattheendof thetextintheformofanappendixtothebook.Amoreextensive coverageofnaivesettheoryisofferedinthebook AxiomsandSet Theory bythisauthor.Itishighlyrecommendedandwillserveas anexcellentcompaniontothisbook.
Asweallknow,anytextbook,wheninitiallypublished,willcontainsomeerrors,sometypographical,othersinspellingorinformattingand,whatisevenmoreworrisome,somemathematical.Critical oralertreadersofthetextcanhelpweedoutthemostglaringmistakesbycommunicatingsuggestionsandcommentsdirectlytothe author.Thiswillbemuchappreciatedbythiswriteraswellasby futurereaders.
RobertAndr´ e UniversityofWaterloo,Ontario
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AbouttheAuthor
RobertAndr´ e isaPh.D.graduatefromtheUniversityofManitoba withaspecializationinPoint-Settopologyunderthesupervisionof Dr.R.GrantWoods.HehaslecturedattheUniversityofManitoba andtheUniversityofReginaand17yearsasafacultymemberofthe DepartmentofPureMathematicsattheUniversityofWaterlooin Ontario,Canada.Healsolecturedmathematicsfor13yearsatthe highschoollevelinManitoba,Saskatchewan,andNewBrunswick.
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Preface vii
AbouttheAuthor xi
PartINormsandMetrics1
Chapter1.NormsonVectorSpaces3
Chapter2.MetricsonSets23
PartIITopologicalSpaces:Fundamental Concepts37
Chapter3.ATopologyonaSet39
Chapter4.SetClosures,InteriorsandBoundaries61
Chapter5.BasesofTopologicalSpaces85
Chapter6.ContinuityonTopologicalSpaces117
Chapter7.ProductSpaces143
Chapter8.TheQuotientTopology189
xiv Point-SetTopologywithTopics
PartIIITopologicalSpaces:SeparationAxioms207
Chapter9.SeparationwithOpenSets209
Chapter10.SeparationwithContinuousFunctions237
PartIVLimitPointsinTopologicalSpaces275
Chapter11.LimitPointsinFirstCountableSpaces277
Chapter12.LimitPointsofNets285
Chapter13.LimitPointsofFilters309 PartVCompactSpacesandRelatives335
Chapter14.Compactness:DefinitionandBasicProperties337
Chapter15.CountablyCompactSpaces363
Chapter16.Lindel¨ofSpaces377
Chapter17.SequentiallyandFeeblyCompactSpaces387
Chapter18.LocallyCompactSpaces407
Chapter19.ParacompactTopologicalSpaces423
PartVITheConnectedProperty439
Chapter20.ConnectedSpacesandProperties441 PartVIITopics475
Chapter21.CompactificationsofCompletely RegularSpaces477
Chapter22.SingularSetsandSingularCompactifications515
Chapter23.On C -EmbeddingsandPseudocompactness537
Chapter24.RealcompactSpaces561
Chapter25.PerfectFunctions581
Chapter26.PerfectandFreudenthalCompactifications597
Chapter27.SpacesWhoseElementsAreSequences611
Chapter28.CompletingIncompleteMetricSpaces633
Chapter29.TheUniformSpaceandtheUniform Topology643
Chapter30.TheStone–WeierstrassTheorem669
Chapter31.Metrizability679
Chapter32.TheStoneSpace693
Chapter33.BaireSpaces723
Chapter34.TheClassof F -Spaces735
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NormsonVectorSpaces
Abstract
Inthissection,wereviewafewbasicnotionsaboutinnerproductson vectorspacesandhowtheyareusedasamechanismtoconstructa distancemeasuringtoolcalled“norm”Wethendefine“normonan abstractvectorspace”withnoreferencetoaninnerproduct.Weshow howtodistinguishbetweennormsthatareinducedbyaninnerproductandthosethatarenot.Wethenprovideafewexamplesofnorms onvectorsspacesofcontinuousfunctions, C [a,b].Weendthissectionwithaformaldefinitionofthecompactpropertyinnormedvector spaces.
1.1ReviewofInnerProductSpaces
Webeginbyreviewingafewbasicnotionsaboutthosevector spaceswerefertoas innerproductspaces.Recallthattheset Rn = {(x1 ,x2 ,x3 ,...,xn ): xi ∈ R} equippedwithtwooperations, additionandscalarmultiplication,isknowntobea vectorspace.We canalsoequipavectorspacewithathirdoperationcalled“dotproduct”whichmapspairsofvectorsin Rn tosomerealnumber.The dot-productisaspecificreal-valuedoperationon Rn whichbelongs toalargerfamilyofvectorspaceoperationscalled innerproducts Webrieflyremindourselvesofafewfactsaboutinnerproductson abstractvectorspaces.
Definition1.1 Let V beavectorspaceoverthereals.An inner product isanoperationwhichmapspairsofvectorsin V toareal number.Wedenoteaninnerproducton V by v, w .Areal-valued functionon V × V isreferredtoasan innerproduct ifandonlyifit satisfiesthefollowingfouraxioms:
IP1: Thenumber v, v isgreaterthanorequalto0forall v in V . Equalityholdsifandonlyif v =0.(Hence,if v isnot0, v, v is strictly largerthan0.)
IP2: Forall v, w ∈ V , u, v = v, u (Commutativity)
IP3: Forall u, v, w ∈ V , u + w, v = u, v + w, v
IP4: Forall u, v ∈ V ,and α ∈ R, αu, v = α u, v
Avectorspace V iscalledan innerproductspace ifitisequipped withsomespecifiedinnerproduct.
Definition1.2 Let V beaninnerproductspace.If v isavectorin V wedefine v = v, v
Theexpression v iscalledthe norm (orlength)ofthevector v inducedbytheinnerproduct u, v on V .
Example1. Itiseasilyverifiedthat,for x =(x1 ,x2 ,x3 ,...,xn )and y =(y1 ,y2 ,y3 ,...,yn )in Rn ,itswell-knowndot-product x, y = x y = x1 y1 + x2 y2 + + xn yn
satisfiesthefouraxiomsaboveandsoisaninnerproductonthe vectorspace Rn .Thisisoftenreferredtoasthe Euclideaninner product orthe standardinnerproduct on Rn .Thenormon Rn induced
bythedotproductis
x = (x1 ,x2 ,x3 ,...,xn ) = (x1 ,x2 ,x3 ,...,xn ) · (x1 ,x2 ,x3 ,...,xn ) = n i=1 x2 i
Thisparticularnormisreferredtoasthe Euclideannorm on Rn . Itisalsoreferredtoasthe L2 -normon Rn ,inwhichcase,itwill berepresentedas x 2 .Wewillusethisparticularnormtomeasure distancesbetweenvectorsin Rn .Thatis,thedistancebetween x = (x1 ,x2 ,x3 ,...,xn )and y =(y1 ,y2 ,y3 ,...,yn )isdefinedtobe x y = n i=1 (xi yi )2
Inthecasewhere n =2or3,thisrepresentstheusualdistance formulabetweenpointsin2-spaceand3-space,respectively.Inthe casewhere n =1,itrepresentstheabsolutevalueofthedifference oftwonumbers.
Example2. Considerthevectorspace, V = C [a,b],thesetofall continuousreal-valuedfunctionsontheclosedinterval[a,b]equipped withtheusualadditionandscalarmultiplicationoffunctions.We definethefollowinginnerproducton C [a,b]as:
= b a
(x)g (x) dx
ShowingthatthisoperationsatisfiestheinnerproductaxiomsIP1 toIP4isleftasanexercise.Inthiscase,thenormof f ,inducedby thisinnerproduct,isseentobe f = b a f (x)2 dx
Itisalsoreferredtoasthe L2 -normon C [a,b]andwerepresentitas f 2 .
Thefollowingtheoremcalledthe Cauchy–Schwarzinequality offersanimportantproblemsolvingtoolwhenworkingwithinner productspaces.Thenormwhichappearsintheinequalityistheone inducedbythegiveninnerproduct.Thisinequalityholdstruefor anywell-definedinnerproductonavectorspace.
Theorem1.3(Cauchy–Schwarzinequality). Let V beavector spaceequippedwithaninnerproductanditsinducednorm.Then, forvectors x and y in V, | x, y |≤ x y
Equalityholdstrueifandonlyif x and y arecollinear(i.e., x = αy or y = αx).
Proof. Thestatementclearlyholdstrueif y =0. Let x, y betwo(notnecessarilydistinct)vectorsin V where y =0. Foranyrealnumber t, 0 ≤ x ty, x ty = x 2 2t x, y + t2 y 2
Choosing t = x, y y 2 intheaboveequationweobtain 0 ≤ x 2 x, y 2 y 2
Theinequality, | x, y |≤ x y ,follows.
Wenowprovethesecondpartofthestatement.If x and y are collinear,say x = αy ,then | x, y | = | αy, y | = |α| y, y = |α| y y = αy y = x y
Conversely,suppose | x, y | = x y .If y iszerothen 0= ty and so x and y arecollinear.Suppose y = 0.Consider t = x, y 2 y 2 . x ty, x ty = x 2 x, y 2 y 2 (asdescribedabove) =0
So,byIP1, x ty, x ty =0implies x ty =0so x and y are collinear.Wehaveshownthatequalityholdsifandonlyif x and y arecollinear.
Wenowverifythatanorm, ,whichisinducedbyaninnerproduct , onthevectorspace V willalwayssatisfythefollowingthree fundamentalproperties:
(1)Forall x ∈ V , x ≥ 0,equalityholdsifandonlyif x =0.
(2)Forall x ∈ V andscalar α ∈ R, αx = |α| x .
(3)Forall x, y ∈ V , x + y ≤ x + y .
Thefirstpropertyfollowsfromthefactthatthenormisdefinedas beingthesquarerootofanumber.Thesecondpropertyfollowsfrom thestraightforwardargument: αx = αx,αx = √α2 x, x = |α| x
Thethirdpropertyisreferredtoasthe triangleinequality.It’snontrivialproofinvokestheCauchy–Schwarzinequality.Weproveit below.
Corollary1.4 Thetriangleinequalityfornormsinducedbyinner products. Foranypairofvectors x and y inaninnerproductspace, x + y ≤ x + y .Ifequalityholdsthenthetwovectors x and y arecollinear.
Proof.
(Cauchy–Schwarz) =( x + y )2
Thus x + y ≤ x + y ,asrequired. Inthecasewherewehaveequality:Supposewehave x + y = x + y .Then x + y 2 =( x + y )2 .Fromthedevelopmentabove,theinequalitiesmustbeequalities,sowemusthave x, y = x y .Thismeans | x, y | = x ||y .BytheCauchy–Schwarztheorem, x and y arecollinear.
Example3. Usethethedotproductpropertiesandthelawof cosines, c2 = a2 + b2 2ab cos θ (where a,b,c representthelengths ofthesidesofatriangle ABC and θ istheanglebetweentothe sides AB and AC )toprovetheCauchy–Schwarzinequalityin R2 .
Solution: Considerthetwovectors a and b in R2 whereoneisnota scalarmultipleoftheother.Thentheextremitiesofthetwovectors formatrianglewithangle θ between a and b.Thenthenorms a , b and b a arenumberswhichcanrepresentthelengthofthesides ofthetriangle.Let θ representtheanglebetween a and b.
Weapplythecosinelawtoobtain b a 2 = a 2 + b 2 2 a b cos θ
If a, b representsthedot-product,weobtain,byapplyingthedotproductpropertiesandtheproperty a 2 = a, a
Example4. Suppose a and b =(1, 0)aretwovectorsin R2 where a =1withanglebetween a and b = θ .Showthat a, b =cos θ ,as isrepresentedinFigure1.1.
Figure1.1. Dot-productintheplane.
Solution: Thisfollowsfromtheaboveexampleand cos θ = a, b a b = a, b 1 × 1
1.2NormsonArbitraryVectorSpaces
Wenowshowthatnormsonavectorspacecanexistindependently frominnerproducts.
Definition1.5 Let V avectorspaceoverthereals.Anormon V isafunction, : V → R,whichsatisfiesthethreefollowingnorm axioms:
N1: Forall v ∈ V , v ≥ 0.Theequality v =0holdstrueifand onlyif v =0.Hence,if v isnot 0, v > 0.
N2: Forall v ∈ V , α ∈ R, αv = |α| v
N3: Forall v, u ∈ V , u + v ≤ u + v (Triangleinequality).
Avectorspace V iscalleda normedvectorspace ifitisequipped withsomespecifiednormsatisfyingthesethreenormaxioms.Itis denotedby(V, ).Notethatthevectorspace V neednotbean innerproductspace,norneedtherebeanyrelationshipbetween andsomeinnerproduct.
Thisfunction, : V → R,willbeusedtomeasurethe“distance”, x y ,betweenanytwovectors x and y .Notethat x y = ( 1)(y x) = |− 1| y x = y x .Thatis,“thedistance between x and y isthesameasthedistancebetween y and x ”. Weprovideafewexamplesoffunctionsknowntobenorms.
Example5. Thenorm inducedbytheinnerproductofaninner productspace(V, , )hasalreadybeenshowntosatisfytheaxioms N1,N2,andN3.
Example6. The1-normon Rn isdefinedas x 1 = (x1 ,x2 ,x3 ,...,xn ) 1 = n i=1 |xi |
Itcanbeshowntosatisfythethreenormaxioms.(Leftasan exercise.)
Example7. The ∞-normon Rn isdefinedas x ∞ = (x1 ,x2 ,x3 ,...,xn ) ∞ =max{|x1 |, |x2 |,..., |xn |}
Itcanbeshowntosatisfythethreenormaxioms.(ProvingthatN1 andN2aresatisfiedisleftasanexercise.)
Weprovethatthe ∞-normon Rn satisfiesthetriangleinequality N3.
Proofof x + y ∞ ≤ x ∞ + y ∞ . Notethat |xi |≤ max i=1...n {|xi |} foreach i,and |yi |≤ maxi=1...n {|yi |} foreach i.Then foreach i =1to n, |
Thenmax 1...n {|xi yi |}≤ x ∞ + y ∞ andso x + y ∞ ≤ x ∞ + y ∞ , asrequired.
Example8. Let p ≥ 1.The p-normon Rn isdefinedas
Itcanbeshownthatthisfunctionsatisfiesthethreenormaxioms. Wewillnotprovethisfor p ingeneral.
Notethatwhen p =2the p-normissimplytheEuclideannorm on Rn .SincetheEuclideannormisinducedbyaninnerproduct,it automaticallysatisfiesthethreenormaxioms.Provingthat,forany p ≥ 1,the p-normsatisfiesN1andN2isstraightforward.Butproving thatthetriangleinequalityholdstrueforall p ≥ 1isnoteasy.The interestedreaderscanlooktheproofupinmostrealanalysistexts orfinditonline.
Anaturalquestioncomestomind .Areallnormsonavector space V inducedbysomeinnerproduct?Theanswerisno!
Thefollowingtheoremtellsushowtorecognizethosenormswhich areinducedbysomeinnerproduct.
Theorem1.6 Suppose isanormonavectorspace V .There existsaninnerproduct , suchthat x = x, x , forall x ∈ V, ifandonlyif, forall x, y ∈ V, thenorm satisfiestheparallelogram identity
Proof. Theproofinvolvesshowingthat,if satisfiestheparallelogramidentitythentheidentity
isavalidinnerproducton V whichinduces .Theproofisroutine andsoisnotpresentedhere.
Examplesofnormson C [a,b]
Example9. The Lp -norm.Let V = C [a,b]denotethefamilyofall real-valuedcontinuousfunctionsontheclosedinterval[a,b]equipped withtheusual+andscalarmultiplication.If p ≥ 1,thenwedefine the Lp -normon C [a,b]asfollows:
Inthecasewhere p =1,wehave
whichiseasilyshowntosatisfythethreenormaxioms.Inthecase where p =2,thenthisnormisonewhichisinducedbytheinner product
on C [a,b].
Itisstraightforwardtoshowthat,forall p ≥ 1, f p satisfiesN1 andN2.Butprovingthat f + g p ≤ f p + g p ,forall p ≥ 1,is difficult.Thisinequalityisreferredtoasthe Minkowskiinequality. Theinterestedreaderswillfindaproofinmostrealanalysistextsor online.
Example10. Thesup-norm:Wedefineanothernormonthevector space, C ∗ (S ),ofallboundedcontinuousreal-valuedfunctionsonthe space S .Recallthatthe supremum ofasubset A ofanorderedset S , writtenas“sup A”isthe leastupperbound of A whichiscontained in S .Notethatsup A mayormaynotbelongto A.Wedefinethe sup-normon C ∗ (S )as
f ∞ =sup {|f (x)| : x ∈ S }
Thesup-normisalsoreferredtoasthe“infinity-norm”or“uniform norm”.Showingthatthisisavalidnormisleftasanexercise.
1.3ConvergenceandCompletenessinaNormed VectorSpace
Convergenceofsequencesformsandimportantpartofanalysis.In whatfollows,normson Rn arealwaysassumedtobetheEuclidean norm,unlessotherwisestated.Aninfinitesequenceinanormedvectorspace, V ,isafunctionwhichmaps N (notnecessarilyone-to-one) ontoasubset, S ,of V .Thismeansthatasequenceistheindexationof theelementsofacountablesubset, S ,byusingthenaturalnumbers. Forexample, S = {x0 , x1 , x2 ,..., xn ,...}.Notethatsomeofthese mayberepeated(sincethefunctionmapping N into S neednotbe one-to-one).Wegeneralizethenotionofa“convergentsequenceand itslimit”in R toa“convergentsequenceanditslimit”inanormed vectorspace(V, ).
Definition1.7 Let(V, +,α, )beavectorspaceequippedwith anorm .Wesayasequenceofvectors {xn } in V convergestoa vector a in V withrespecttothenorm if
Forany ε> 0,thereexistsaninteger N> 0suchthat xn a|| <ε whenever n>N .
