Undergraduate Topology

by Robert H. Kasriel

Overview

General topology offers a valuable tool to students of mathematics, particularly in courses involving complex, real, and functional analysis. This introductory treatment is essentially self-contained, and it features explanations and proofs that relate to every practical aspect of point-set topology. It will prove valuable to undergraduate mathematics majors as well as to graduate students and professionals pursuing mathematics research.
Author Robert H. Kasriel, who taught at Georgia Tech for many years, begins with reviews of elementary set theory and Euclidean n-space. The following chapters offer detailed studies of metric spaces and applications to analysis. A survey of general topological spaces and mappings includes considerations of compactness, connectedness, quotient spaces, net and filter convergence, and product spaces. Nearly every one of the 112 sections in this book concludes with a set of exercises that reinforce materials already covered and prepare students for subsequent chapters.

Product Details

ISBN-13:	9780486474199
Publisher:	Dover Publications
Publication date:	10/22/2009
Series:	Dover Books on Mathematics
Pages:	304
Product dimensions:	6.10(w) x 9.20(h) x 0.80(d)

About the Author

A specialist in topology, Robert H. Kasriel (1918–2007) was a Professor of Mathematics at Georgia Tech for more than 30 years.

Table of Contents

1 Sets, Functions, and Relations 1

1 Sets and Membership 1

2 Some Remarks on the Use of the Connectives and, or, implies 3

3 Subsets 7

4 Union and Intersection of Sets 7

5 Complementation 8

6 Set Identities and Other Set Relations 9

7 Counterexamples 11

8 Collections of Sets 12

9 Cartesian Product 14

10 Functions 15

11 Relations 18

12 Set Inclusions for Image and Inverse Image Seta 23

13 The Restriction of a Function 26

14 Composition of Functions 27

15 Sequences 30

16 Subsequences 33

17 Finite Induction and Well-Ordering for Positive Integers 34

18 Sequencea Defined Inductively 35

19 Some Important Properties of Relations 38

20 Decomposition of a Set 40

21 Equivalence Classes 41

22 Partially Ordered and Totally Ordered Seta 42

23 Properties of Boundedness for Partially Ordered Sets 43

24 Axiom of Choice and Zorn's Lemma 46

25 Cardinality of Sets (Introduction) 48

26 Countable Sets 49

27 Uncountable Sets 51

28 Nonequivalent Sets 53

29 Review Exercises 54

2 Structure Of R and Rn 58

30 Algebraic Structure of R 59

31 Distance Between Two Points in R 63

32 Limit of a Sequence in R 63

33 The Nested Interval Theorem for R 64

34 Algebraic Structure for Rⁿ 65

35 The Cauchy-Schwarz Inequality 67

36 The Distance Formula in Rⁿ 68

37 Open Subsets of Rⁿ 70

38 Limit Points in Rⁿ 71

39 Closed Subsets of Rⁿ 72

40 Bounded Subsets of R 73

41 Convergent Sequences in Rⁿ 75

42 Cauchy Criterion for Convergence 78

43 Some Additional Properties for Rⁿ 79

44 Some Further Remarks about Rⁿ 80

3 Metric Spaces: Introduction 81

45Distance Function and Metric Spaces 81

46 Open Sets and Closed Sets 83

47 Some Basic Theorems Concerning Open and Closed Sets 85

48 Topology Generated by a Metric 86

49 Subspace of a Metric Space 87

50 Convergent Sequences in Metric Spaces 89

51 Cartesian Product of a Finite Number of Metric Spaces 90

52 Continuous Mappings: Introduction 94

53 Uniform Continuity 103

4 Metric Spaces: Special Properties and Mappings on Metric Spaces 107

54 Separation Properties 108

55 Connectedness in Metric Spaces 110

56 The Invariance of Connectedness under Continuous Mappings 113

57 Polygonal Connectedness 114

58 Separable Metric Spaces 116

59 Totally Bounded Metric Spaces 118

60 Sequential Compactness for Metric Spaces 121

61 The Bolzano-Weierstrass Property 123

62 Compactness or Finite Subcovering Property 124

63 Complete Metric Spaces 127

64 Nested Sequences of Sets for Complete Spaces 128

65 Another Characterization of Compact Metric Spaces 131

66 Completion of a Metric Space 131

67 Sequences of Mappings into a Metric Space 136

68 Review Exercises 140

5 Metric Spaces: Some Examples and Applications 143

69 Linear or Vector Spaces 144

70 The Hilbert Space l² 146

71 The Hilbert Cube 151

72 The Space & ([a, b]) of Continuous Real-Valued Mappings on a Closed Interval [a, b] 153

73 An Application of Completeness: Contraction Mappings 156

74 Fundamental Existence Theorem for First Order Differential Equations-An Application of the Banach Fixed Point Theorem 159

6 General Topological Spaces and Mappings on Topological Spaces 162

75 Topological Spaces 163

76 Base for a Topology 165

77 Some Basic Definitions 170

78 Some Basic Theorems for Topological Spaces 173

79 Neighborhoods and Neighborhood Systems 176

80 Subspaces 177

81 Continuous and Topological Mappings 179

82 Some Basic Theorems Concerning Mappings 180

83 Separation Properties for Topological Spaces 183

84 A Characterization of Normality 187

85 Separability Axiom 193

86 Second Countable Spaces 194

87 First Countable Spaces 196

88 Comparison of Topologies 197

89 Urysohn's Metrization Theorem 198

7 Compactness And Related Properties 202

90 Definitions of Various Compactness Properties 203

91 Some Consequences of Compactness 205

92 Relations Between Various Types of Compactness 207

93 Local Compactness 211

94 The One-Point Compactification 213

95 Some Generalizations of Mappings Denned on Compact Spaces 214

8 Connectedness And Related Concepts 218

96 Connectedness. Definitions 219

97 Some Basic Theorems Concerning Connectedness 221

98 Limit Superior and Limit Inferior of Sequences of Subsets of a Space 225

99 Review Exercises 226

9 Quotient Spaces 230

100 Decomposition of a Topological Space 231

101 Quasi-Compact Mappings 233

102 The Quotient Topology 235

103 Decomposition of a Domain Space into Point Inverses 239

104 Topologically Equivalent Mappings 242

105 Decomposition of a Domain Space into Components of Point Inverses 243

106 Factorization of Compact Mappings 243

10 Net and Filter Convergence 248

107 Nets and Subnets 248

108 Convergence of Nets 251

109 Filters 255

11 Product Spaces 261

110 Cartesian Products 261

111 The Product Topology 262

112 Mappings into Product Spaces 269

References 277

Index 279