Basic Topology 1: Metric Spaces and General Topology

523 Pages·2022·5.107 MB·English

by  Avishek Adhikari, Mahima Ranjan Adhikari

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
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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
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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