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1、COMPANIONNOTESAWorkingExcursiontoAccompanyBabyRudinEvelynM.Silvia1April1,19991hcEvelynM.Silvia1999iiContentsPrefacevii0.0.1AbouttheOrganizationoftheMaterial...........viii0.0.2AbouttheErrors.......................viii1TheFieldofRealsandBeyond11.1Fields.................................11.2Ordere
2、dFields............................111.2.1SpecialSubsetsofanOrderedField.............151.2.2BoundingProperties.....................171.3TheRealField.............................251.3.1DensityPropertiesoftheReals................271.3.2ExistenceofnthRoots....................291.3.3TheExtendedReal
3、NumberSystem.............331.4TheComplexField..........................341.4.1ThinkingComplex......................391.5ProblemSetA.............................432FromFinitetoUncountableSets492.1SomeReviewofFunctions......................492.2AReviewofCardinalEquivalence..................542.2.
4、1DenumerableSetsandSequences..............612.3ReviewofIndexedFamiliesofSets.................622.4CardinalityofUnionsOverFamilies.................652.5TheUncountableReals........................692.6ProblemSetB.............................703METRICSPACESandSOMEBASICTOPOLOGY733.1Euclideann-space
5、...........................73iiiivCONTENTS3.2MetricSpaces.............................773.3PointSetTopologyonMetricSpaces.................823.3.1ComplementsandFamiliesofSubsetsofMetricSpaces..873.3.2OpenRelativetoSubsetsofMetricSpaces.........963.3.3CompactSets.........................983.3.4Co
6、mpactnessinEuclideann-space..............1053.3.5ConnectedSets........................1113.3.6PerfectSets..........................1143.4ProblemSetC.............................1184SequencesandSeries–FirstView1234.1SequencesandSubsequencesinMetricSpaces...........1244.2CauchySequencesinMetricSp
7、aces.................1354.3SequencesinEuclideank-space...................1394.3.1UpperandLowerBounds..................1454.4SomeSpecialSequences.......................1494.5SeriesofComplexNumbers.....................1524.5.1Some(A
