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1、COMPANIONNOTESAWorkingExcursiontoAccompanyBabyRudinEvelynM.Silvia1April1,19991hcEvelynM.Silvia1999iiContentsPrefacevii0.0.1AbouttheOrganizationoftheMaterial...........viii0.0.2AbouttheErrors.......................viii1TheFieldofRealsandBeyond11.1Fields.................................11.2O
2、rderedFields............................111.2.1SpecialSubsetsofanOrderedField.............151.2.2BoundingProperties.....................171.3TheRealField.............................251.3.1DensityPropertiesoftheReals................271.3.2ExistenceofnthRoots....................291.3.3TheEx
3、tendedRealNumberSystem.............331.4TheComplexField..........................341.4.1ThinkingComplex......................391.5ProblemSetA.............................432FromFinitetoUncountableSets492.1SomeReviewofFunctions......................492.2AReviewofCardinalEquivalence.........
4、.........542.2.1DenumerableSetsandSequences..............612.3ReviewofIndexedFamiliesofSets.................622.4CardinalityofUnionsOverFamilies.................652.5TheUncountableReals........................692.6ProblemSetB.............................703METRICSPACESandSOMEBASICTOPOLOGY7
5、33.1Euclideann-space...........................73iiiivCONTENTS3.2MetricSpaces.............................773.3PointSetTopologyonMetricSpaces.................823.3.1ComplementsandFamiliesofSubsetsofMetricSpaces..873.3.2OpenRelativetoSubsetsofMetricSpaces.........963.3.3CompactSets.........
6、................983.3.4CompactnessinEuclideann-space..............1053.3.5ConnectedSets........................1113.3.6PerfectSets..........................1143.4ProblemSetC.............................1184SequencesandSeries–FirstView1234.1SequencesandSubsequencesinMetricSpaces...........1
7、244.2CauchySequencesinMetricSpaces.................1354.3SequencesinEuclideank-space...................1394.3.1UpperandLowerBounds..................1454.4SomeSpecialSequences.......................1494.5SeriesofComplexNumbers.....................1524.5.1Some(A
