Basic Topology 1: Metric Spaces and General Topology PDF
Preview Basic Topology 1: Metric Spaces and General Topology
Avishek Adhikari Mahima Ranjan Adhikari Basic Topology 1 Metric Spaces and General Topology Basic Topology 1 ProfessorMahimaRanjanAdhikari(1944–2021) · Avishek Adhikari Mahima Ranjan Adhikari Basic Topology 1 Metric Spaces and General Topology AvishekAdhikari MahimaRanjanAdhikari DepartmentofMathematics InstituteforMathematics,Bioinformatics, PresidencyUniversity InformationTechnologyandComputer Kolkata,WestBengal,India Science(IMBIC) Kolkata,WestBengal,India ProfessorMahimaRanjanAdhikariisdeceased ISBN978-981-16-6508-0 ISBN978-981-16-6509-7 (eBook) https://doi.org/10.1007/978-981-16-6509-7 MathematicsSubjectClassification:54-XX,46A11,18-XX,46Axx,46Bxx ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SingaporePteLtd.2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSingaporePteLtd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Dedicatedto mygrandparentsNabaKumarandSnehalataAdhikari, mymotherMinatiAdhikari whocreatedmyinterestinmathematics atmyveryearlychildhood. AvishekAdhikari Dedicatedto Prof.A.C.ChaudhuriandProf.B.C.Chatterjee whogavemethefirstlessonoftopology attheUniversityofCalcutta, India, inthesession1965–1966. MahimaRanjanAdhikari Preface This volume studies metric spaces and general topology. It considers the general propertiesoftopologicalspacesandtheirmappings.Thespecialstructureofametric spaceinducesatopologyhavingmanyapplicationsoftopologyinmodernanalysis, geometry and algebra. Contents of Volume 1 are expanded in eight chapters. The chapterwisetextsrunasfollows: Chapter1assemblestogethersomebasicconceptsandresultsofsettheory,alge- braicsystems,analysis,Euclideanspacesandcategorytheorythroughtheconcepts of categories, functors and natural transformations with the standard notations for smoothreadingofthebook. Chapter2startswiththeconceptofthemetrics,whichisanabstractionofdistance intheEuclideanspaceandconveysanaxiomaticframeworkforthisabstractionwitha systemicstudyofelementarybasicpropertiesofmetricspaceswithaviewtodefine open sets and hence to study continuity of functions. Urysohn lemma for metric spaces facilitates to provide a vast number of continuous functions while metric spacesprovidearichsupplyoftopologicalspaces.Infact,mostoftheapplications of topology to analysis arise through metric spaces. Normed linear spaces form a special class of metric spaces which provide Banach and Hilbert spaces. Metric spacesgivethesimplestsettingforthestudyofcertainproblemsarisinginanalysis. Theframeworkfortopologybeginswithanintroductiontometricspaces.Thespecial structureofametricspaceinducesatopologyhavingmanyapplicationsoftopology inmodernanalysisandmodernalgebra. Chapter3conveysthebasicconceptsoftopologicalspaces.Infact,theircontin- uousmappingsinanaxiomaticframeworkbyintroducingtheconceptofopensets withoutthenotionofdistancefunctionormetric,wherethebasicobjectsaretopo- logical spaces and basic functions between them are continuous maps. The basic motivationofframingtheaxiomsistointroducethenotionofcontinuity,whichis the central concept in topology. It provides a convenient language to study when differentpointsinaspacecomeneartoeachother,andhence,thesubjectplaysan importantroleinscienceandtechnology. Chapter4studiestopologicalspacesbyimposingcertainconditions,calledsepa- ration axioms, as the defining properties of a topological are weak to study most vii viii Preface of the topological spaces of our interest, which carry more structure (not like a metric).TheseaxiomsinitiallyusedbyP.S.Alexandroff(1896–1982)andH.Hopf (1894–1971)facilitatetoclassifytopologicalspacesandprovideenoughsupplyof continuous functions which are linked to open sets. Separation properties provide enoughsupplyofcontinuousfunctionswhicharelinkedtoopensets.Manyimpor- tanttopologicalpropertiescanbecharacterizedwiththehelpofseparationaxioms by distributing the open sets in the space X and imposing natural conditions on X suchthatX behaveslikeametricspace. Chapter5studiescompactnessandconnectednessintopologicalsettings,which aretwoimportanttopologicalproperties.Acompactspaceisanaturalgeneralization ofclosedandboundedsetsintheEuclideanspaceRn.Ontheotherhand,theconcept ofconnectednessasasinglepiecegeneralizestheintuitiveideaofnonseparatenessof ageometricobject.Thesetwotopologicalconceptsareutilizedtosolvemanyprob- lemsintopology,mainlyclassificationoftopologicalspacesuptohomeomorphism, and are fundamental in the study of modern analysis, geometry, topology, algebra and many other areas. Moreover, this chapter studies compactification, which is a process or result of making a topological space into a compact space. There are many noncompact spaces. Considering the importance of compactness in mathe- matics,thisstudyincludesStone–CˇechcompactificationandAlexandroffone-point compactification. Chapter6presentsmoreresultsoncontinuousfunctionsfromatopologicalspace tothereallinespace,calledreal-valuedcontinuousfunctions,whichplaysacentral roleintopologyandstudiesuniformconvergenceandnormalspacesthroughsepara- tionbysuchfunctions.ThischapterprovesUrysohnlemmafornormalspaceswith the help of dyadic rational numbers, which is a surprising result and applies it to Tietzeextensiontheoremandringtheory. Chapter7studiescertainclassoftopologicalspacessatisfyingthetwoaxiomsof countabilityformulatedbyF.Hausdorffin1914andestablishesaconnectionbetween compactnessandtheBolzano–Weierstrassproperty(B-Wproperty)andprovessome embedding theorem. Motivation for the study of the concepts of countability and separabilityoftopologicalspacescomesfromsomenaturalproblemsdiscussedin thischapter. Chapter 8 conveys the history of emergence of the concepts leading to the developmentofgeneraltopologyasasubjectwiththeirmotivations. The book is a clear exposition of the basic ideas of topology and conveys a straightforward discussion of the basic topics of topology and avoids unnecessary definitionsandterminologies.Eachchapterstartswithhighlightingthemainresults ofthechapterwithmotivationandissplitintoseveralsectionswhichdiscussrelated topicswithsomedegreeofthoroughnessandendswithexercisesofvaryingdegrees ofdifficulties,whichnotonlyimpartanadditionalinformationaboutthetextcovered previouslybutalsointroduceavarietyofideasnottreatedintheearliertextswith certain references to the interested readers for more study. All these constitute the basicorganizationalunitsofthebook. Thisthree-volumebooktogetherwiththeauthors’twootherSpringerbooksBasic ModernAlgebrawithApplications(Springer,2014)andBasicAlgebraicTopology Preface ix and its Applications (Springer, 2016) will form a unitary module for the study of modernalgebra,generalandalgebraictopologywithapplicationsinseveralareas. TheauthorsacknowledgetheHigherEducationDepartmentoftheGovernmentof WestBengalforsanctioningthefinancialsupporttothe“InstituteforMathematics, BioinformaticsandComputerScience(IMBIC)”towardwritingthisbookvideorder no432(Sanc)/EH/P/SE/SE/1G-17/07datedAugust29,2017,andalsotoIMBIC, University of Calcutta, Presidency University, Kolkata, India, and Moulana Abul KalamAzadUniversityofTechnology,WestBengal,forprovidingtheinfrastructure towardimplementingthescheme. The authors are indebted to the authors of the books and research papers listed in the bibliography at the end of each chapter and are very thankful to Profs. P. Stavrions(Greece),ConstantineUdriste(Romania),andAkiraAsada(Japan)andto thereviewersofthemanuscriptfortheirscholarlysuggestionsforimprovements.We arethankfultoMd.KutubuddinSardarforhiscooperationtowardsthetypesetting of the manuscript and to many UG and PG students of Presidency University and CalcuttaUniversity,andtomanyotherindividualswhohavehelpedinproofreading the book. Authors apologize to those whose names have been inadvertently not entered. Finally, the authors acknowledge, with heartfelt thanks, the patience and sacrifice of long-suffering family of the authors, especially Dr. Shibopriya Mitra AdhikariandMasterAvipriyoAdhikari. Kolkata,India AvishekAdhikari June2021 MahimaRanjanAdhikari A Note on Basic Topology—Volumes 1–3 Thetopic“Topology”hasbecomeoneofthemostexcitingandinfluentialfieldsof studyinmodernmathematics,becauseofitsbeautyandscope.Theaimofthissubject istomakeaqualitativestudyofgeometryinthesensethatifonegeometricobject iscontinuouslydeformedintoanothergeometricalobject,thenthesetwogeometric objects are considered topologically equivalent, called homeomorphic. Topology starts where sets have some cohesive properties, leading to define continuity of functions. The series of three books on Basic Topology is a project book funded by the Government of West Bengal, which is designed to introduce many variants of a basiccourseintopologythroughthestudyofpointsettopology,topologicalgroups, topological vector spaces, manifolds, Lie groups, homotopy and homology theo- ries with an emphasis of their applications in modern analysis, geometry, algebra andtheoryofnumbers.Topicsintopologyarevast.Therangeofitsbasictopicsis distributedamongdifferenttopologicalsubfieldssuchasgeneraltopology,topolog- ical algebra, differential topology, combinatorial topology, algebraic topology and geometric topology. Each volume of the present book is considered as a separate textbookthatpromotesactivelearningofthesubjecthighlightingelegance,beauty, scopeandpoweroftopology. BasicTopology—Volume1:MetricSpacesandGeneral Topology This volume majorly studies metric spaces and general topology. It considers the generalpropertiesoftopologicalspacesandtheirmappings.Thespecialstructureof ametricspaceinducesatopologyhavingmanyapplicationsoftopologyinmodern analysis,geometryandalgebra.ThetextsofVolume1areexpandedineightchapters. xi
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