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时间:2018-08-02
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1、COMPANIONNOTESAWorkingExcursiontoAccompanyBabyRudinEvelynM.Silvia1April1,19991hcEvelynM.Silvia1999iiContentsPrefacevii0.0.1AbouttheOrganizationoftheMaterial...........viii0.0.2AbouttheErrors.......................viii1TheFieldofRealsandBeyond11.1Fields...............................
2、..11.2OrderedFields............................111.2.1SpecialSubsetsofanOrderedField.............151.2.2BoundingProperties.....................171.3TheRealField.............................251.3.1DensityPropertiesoftheReals................271.3.2ExistenceofnthRoots..................
3、..291.3.3TheExtendedRealNumberSystem.............331.4TheComplexField..........................341.4.1ThinkingComplex......................391.5ProblemSetA.............................432FromFinitetoUncountableSets492.1SomeReviewofFunctions......................492.2AReviewofCardina
4、lEquivalence..................542.2.1DenumerableSetsandSequences..............612.3ReviewofIndexedFamiliesofSets.................622.4CardinalityofUnionsOverFamilies.................652.5TheUncountableReals........................692.6ProblemSetB.............................703METRI
5、CSPACESandSOMEBASICTOPOLOGY733.1Euclideann-space...........................73iiiivCONTENTS3.2MetricSpaces.............................773.3PointSetTopologyonMetricSpaces.................823.3.1ComplementsandFamiliesofSubsetsofMetricSpaces..873.3.2OpenRelativetoSubsetsofMetricSpaces.
6、........963.3.3CompactSets.........................983.3.4CompactnessinEuclideann-space..............1053.3.5ConnectedSets........................1113.3.6PerfectSets..........................1143.4ProblemSetC.............................1184SequencesandSeries–FirstView1234.1Sequence
7、sandSubsequencesinMetricSpaces...........1244.2CauchySequencesinMetricSpaces.................1354.3SequencesinEuclideank-space...................1394.3.1UpperandLowerBounds..................1454.4SomeSpecialSequences.......................1494.5SeriesofComplexNumbers................
8、.....1524.5.1Some(A
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