普通拓扑学

CHAPTERO:PRELIMINARIES
SETS
SUBSETSANDCOMPLEMENTS;UNIONANDINTERSECTION
Classcalculus
RELATIONS
Relationcalculus,equivalencerelations
FUNCTIONS
ORDERINGS
Order-completesets,chains,extensionoforder-preservingfunctions
ALGEBRAICCONCEPTS
THEREALNUMBERS
Integers,definitionbyinduction,b-adicexpansionsCOUNTABLESETS
Subsets,unions,thesetofrealnumbersCARDINALNUMBERS
Schroeder-Bernsteintheorem
ORDINALNUMBERS
Thefirstuncountableordinal
CARTESIANPRODUCTS
HAUSDORFFMAXIMALPRINCIPLE
Maximalprinciple,Kuratowski-Zornlemma,axiomofchoice,well-orderingprinciple
CHAPTER1:TOPOLOGICALSPACES
TOPOLOGIESANDNEIGHBORHOODS
Comparisonoftopologies,neighborhoodsystemofapointCLOSEDSETS
ACCUMULATIONPOINTS
CLOSURE
Kuratowskiclosureoperators
INTERIORANDBOUNDARY
BASESANDSUBBASES
Topologieswithacountablebase,Lindel6ftheorem
RELATIVIZATION;SEPARATION
CONNECTEDSETS
Components
PROBLEMS
ALargestandsmallesttopologies;BTopologiesfromneighborhoodsystems;CTopologiesfrominterioroperators;DAccumulationpointsinTi-spaces;EKuratowskiclosureandcomplementproblem;FExerciseonspaceswithacountablebase;GExerciseondensesets;HAccumulationpoints;ITheordertopology;JPropertiesoftherealnumbers;KHalf-openintervalspace;LHalf-openrectanglespace;MExample(theordinals)on1stand2ndcountability;NCountablechaincondition;OTheEuclideanplane;PExampleoncomponents;QTheoremonseparatedsets;RFinitechaintheoremforconnectedsets;SLocallyconnectedspaces;
TTheBrouwerreductiontheorem
CHAPTER2:MOORE-SMITHCONVERGENCE
INTRODUCTION
DIRECTEDSETSANDNETS
Uniquenessoflimits,iteratedlimits
SUBNETSANDCLUSTERPOINTS
SEQUENCESANDSUBSEQUENCES
CONVERGENCECLASSES
Specificationofatopologybyconvergence
PROBLEMS
AExerciseonsequences;BExample:sequencesareinadequate;CExerciseonHausdorffspaces:doorspaces;DExerciseonsubsequences;EExample:cofinalsubsetsareinadequate;FMonotonenets;GIntegrationtheory,juniorgrade;HIntegrationtheory,utilitygrade;IMaximalidealsinlattices;JUniversalnets;KBooleanrings:thereareenoughhomomorphisms;LFilters
CHAPTER3:PRODUCTANDQUOTIENTSPACES
CONTINUOUSFUNCTIONS
Characterizationsofcontinuity,homeomorphismsPRODUCTSPACES
Functionstoaproduct,coordinatewiseconvergence,countability
QUOTIENTSPACES
Openandclosedmaps,uppersemi-continuousdecompositions
PROBLEMS
AConnectedspaces;BTheoremoncontinuity;CExerciseoncontinuousfunctions;DContinuityatapoint;continuousextension;EExerciseonreal-valuedcontinuousfunctions;FUppersemi-continuousfunctions;GExerciseontopologicalequivalence;HHomeomorphismsandone-to-onecontinuousmaps;IContinuityineachoftwovariables;JExerciseonEuclideann-space;KExerciseonclosure,interiorandboundaryinproducts;LExerciseonproductspaces;MProductofspaceswithcountablebases;NExampleonproductsandseparability;OProductofconnectedspaces;PExerciseonT1-spaces;QExerciseonquotientspaces;RExampleonquotientspacesanddiagonalsequences;STopologicalgroups;TSubgroupsofatopologicalgroup;UFactorgroupsandhomomorphisms;VBoxspaces;WFunctionalsonreallinearspaces;XReallineartopologicalspaces
CHAPTER4:EMBEDDINGANDMETRIZATION
EXISTENCEOFCONTINUOUSFUNCTIONS
Tychonofflemma,Urysohnlemma
EMBEDDINGINCUBES
Embeddinglemma,Tychonoffspaces
METRICANDPSEUDO-METRICSPACES
Metrictopology,countableproducts
METRIZATION
Urysohnmetrizationtheorem,locallyfinitecovers,refinement,characterizationofmetrizability
PROBLEMS
ARegularspaces;BContinuityoffunctionsonametricspace;CProblemonmetrics;DHausdorffmetricforsubsets;EExample(theordinals)ontheproductofnormalspaces;FExample(theTychonoffplank)onsubspacesofnormalspaces;GExample:productsofquotientsandnon-regularHausdorffspaces;HHereditary,productive,anddivisibleproperties;IHalf-openintervalspace;JThesetofzerosofarealcontinuousfunction;KPerfectlynormalspaces;LCharacterizationofcompletelyregularspaces;MUppersemi-continuousdecompositionofanormalspace
CHAPTER5:COMPACTSPACES
EQUIVALENCES
Finiteintersectionproperty,clusterpoints,Alexander
subbasetheorem
COMPACTNESSANDSEPARATIONPROPERTIES
CompactnessforHausdorff,regularandcompletelyregularspaces
PRODUCTSOFCOMPACTSPACES
TheTychonoffproducttheorem
LOCALLYCOMPACTSPACES
QUOTIENTSPACES
U.s.c.decompositionswithcompactmembers
COMPACTIFICATION
AlexandroffonepointandStone-Cechcompactifications
LEBESGUE'SCOVERINGLEMMA
Evencoverings
PARACOMPACTNESS
PROBLEMS
AExerciseonrealfunctionsonacompactspace;BCompactsubsets;CCompactnessrelativetotheordertopology;DIsometricsofcompactmetricspaces;ECountablycompactandsequentiallycompactspaces;FCompactness;theintersectionofcompactconnectedsets;GProblemonlocalcompactness;HNestcharacterizationofcompactness;ICompleteaccumulationpoints;JExample:unitsquarewithdictionaryorder;KExample(theordinals)onnormalityandproducts;LThetransfiniteline;MExample:theHellyspace;NExamplesonclosedmapsandlocalcompactness;OCantorspaces;PCharacterization.ofStone-Cechcompactification;QExample(theordinals)oncompactification;RTheWallmancompactification;SBooleanrings:Stonerepresentationtheorem;TCompactconnectedspaces(thechainargument);UFullynormalspaces;VPointfinitecoversandmetacompactspaces;WPartitionofunity;XThebetweentheoremforsemi-continuousfunctions;YParacompactspaces
CHAPTER6:UNIFORMSPLICES
UNIFORMITIESANDTHEUNIFORMTOPOLOGY
Neighborhoods,basesandsubbases
UNIFORMCONTINUITY;PRODUCTUNIFORMITIES
Uniformisomorphism,relativization,products
METRIZATION
Characterizationofmetrizability,thegageofauniformity
COMPLETENESS
Cauchynets,extensionoffunctions
COMPLETION
Existenceanduniqueness
COMPACTSPACES
Uniquenessofuniformity,totalboundedness
FORMETRICSPACESONLY
Bairetheorem,localizationofcategory,uniformlyopen
maps
PROBLEMS
AExerciseonclosedrelations;BExerciseontheproductoftwouniformspaces;CAdiscretenon-metrizableuniformspace;DExercise:uniformspaceswithanestedbase;EExample:averyincompletespace(theordinals);FThesubbasetheoremfortotalboundedness;GSomeextremaluniformities;HUniformneighborhoodsystems;IEcartsandmetrics;JUniformcoveringsystems;KTopologicallycompletespaces:metrizablespaces;LTopologicallycompletespaces:uniformizablespaces;MThediscretesubspaceargument;countablecompactness;NInvariantmetrics;OTopologicalgroups:uniformitiesandmetrization;PAlmostopensubsetsofatopologicalgroup;QCompletionoftopologicalgroups;RContinuityandopennessofhomomorphisms:theclosedgraphtheorem;SSummability;TUniformlylocallycompactspaces;UTheuniformboundednesstheorem;VBoolean-rings
CHAPTER7:FUNCTIONSPACES
POINTWISECONVERGENCE
Topologyanduniformity,compactness
COMPACTOPENTOPOLOGYANDJOINTCONTINUITY
Uniquenessofjointlycontinuoustopologies,C.G.compact
spaces
UNIFORMCONVERGENCE
Uniformconvergenceonafamilyofsets,completeness
UNIFORMCONVERGENCEONCOMPACTA
Topology,completeness,k-spaces
COMPACTNESSANDEQUICONTINUITY
TheAscolitheorem
EVENCONTINUITY
TopologicalAscolitheorem
PROBLEMS
AExerciseonthetopologyofpointwiseconvergence;
BExerciseonconvergenceoffunctions;CPointwiseconvergenceonadensesubset;DThediagonalprocessandsequentialcompactness;EDini'stheorem;FContinuityofaninducedmap;GUniformequicontinuity;HExerciseontheuniformityIContinuityofevaluation;JSubspaces,productsandquotientsofk-spaces;KThek-extensionofatopology;LCharacterizationofevencontinuity;MContinuousconvergence;NTheadjointofanormedlinearspace;OTietzeextensiontheorem;PDensitylemmaforlinearsubspacesofC(X);QThesquarerootlemmaforBanachalgebras;RTheStone-Weierstrasstheorem;SStructureofC(X);TCompactificationofgroups;almostperiodicfunctions
APPENDIX:ELEMENTARYSETTHEORY
CLASSIFICATIONAXIOMSCHEME
Axiomofextentandclassificationaxiomscheme
CLASSIFICATIONAXIOMSCHEME(Continued)
Formalstatementofclassificationaxiomscheme
ELEMENTARYALGEBRAOFCLASSES
EXISTENCEOFSETS
Axiomofsubsets,axiomofunion,unorderedpairs
ORDEREDPAIRS;RELATIONS
FUNCTIONS
Axiomofsubstitution,axiomofamalgamation
WELLORDERING
ExistenceanduniquenessoforderpreservingfunctionsORDINALS
Axiomofregularity,structureofordinals)transfiniteinduction
INTEGERS
Axiomofinfinity,Peanopostulatesforintegers
THECHOICEAXIOM
Themaximalprinciple
CARDINALNUMBERS
Elementaryproperties,finitesets,theproductofcardinalsBIBLIOGRAPHY
INDEX