## Robert H. Kasriel: Undergraduate Topology

Published $\text {1971}$, Dover Publications, Inc.

ISBN 0-486-47419-4.

### Contents

Preface
To the Instructor
Notation for Some Important Sets

1. Sets, Functions, and Relations
1. Sets and Membership
2. Some Remarks on the Use of the Connectives and, or, implies
3. Subsets
4. Union and Intersection of Sets
5. Complementation
6. Set Identities and Other Set Relations
7. Counterexamples
8. Collections of Sets
9. Cartesian Product
10. Functions
11. Relations
12. Set Inclusions for Image and Inverse Image Sets
13. The Restriction of a Function
14. Composition of Functions
15. Sequences
16. Subsequences
17. Finite Induction and Well-Ordering for Positive Integers
18. Sequences Defined Inductively
19. Some Important Properties of Relations
20. Decomposition of a Set
21. Equivalence Classes
22. Partially Ordered and Totally Ordered Sets
23. Properties of Boundedness for Partially Ordered Sets
24. Axiom of Choice and Zorn's Lemma
25. Cardinality of Sets (Introduction)
26. Countable Sets
27. Uncountable Sets
28. Nonequivalent Sets
29. Review Exercises

2. Structure of $\mathbf R$ and $\mathbf R^n$
30. Algebraic Structures of $\mathbf R$
31. Distance Between Two Points in $\mathbf R$
32. Limit of a Sequence in $\mathbf R$
33. The Nested Interval Theorem in $\mathbf R$
34. Algebraic Structure for $\mathbf R^n$
35. The Cauchy-Schwarz Inequality
36. The Distance Forumula in $\mathbf R^n$
37. Open Subsets of $\mathbf R^n$
38. Limit Points in $\mathbf R^n$
39. Closed Subsets of $\mathbf R^n$
40. Bounded Subsets of $\mathbf R^n$
41. Convergent Sequences in $\mathbf R^n$
42. Cauchy Criterion for Convergence
43. Some Additional Properties for $\mathbf R^n$
44. Some Further Remarks About $\mathbf R^n$

3. Metric Spaces: Introduction
45. Distance Function and Metric Spaces
46. Open Sets and Closed Sets
47. Some Basic Theorems Concerning Open and Closed Sets
48. Topology Generated by a Metric
49. Subspace of a Metric Space
50. Convergent Sequences in Metric Spaces
51. Cartesian Product of a Finite Number of Metric Spaces
52. Continuous Mappings: Introduction
53. Uniform Continuity

4. Metric Spaces: Special Properties and Mappings on Metric Spaces
54. Separation Properties
55. Connectednes in Metric Spaces
56. The Invariance of Connectedness Under Continuous Mappings
57. Polygonal Connectedness
58. Separable Metric Spaces
59. Totally Bounded Metric Spaces
60. Sequential Compactness for Metric Spaces
61. The Bolzano-Weierstrass Property
62. Compactness or Finite Subcovering Property
63. Complete Metric Spaces
64. Nested Sequences of Sets for Complete Spaces
65. Another Characterization of Compact Metric Spacs
66. Completion of a Metric Space
67. Sequences of Mappings into a Metric Space
68. Review Exercises

5. Metric Spaces: Some Examples and Applications
69. Linear or Vector Spaces
70. The Hilbert Space $ell^2$
71. The Hilbert Cube
72. The Space $\map {\mathscr C} {\sqbrk {a, b} }$ of Continuous Real-Valued Mappings on a Closed Interval $\sqbrk {a, b}$
73. An Application of Completeness: Contraction Mappings
74. Fundamental Existence Theorem for First Order Differential Equations -- An Application of the Banach Fixed Point Theorem

6. General Topological Spaces and Mappings on Topological Spaces
75. Topological Spaces
76. Base for a Topology
77. Some Basic Definitions
78. Some Basic Theorems for Topological Spaces
79. Neighborhoods and Neighborhood Systems
80. Subspaces
81. Continuous and Topological Mappings
82. Some Basic Theorems Concerning Mappings
83. Separation Properties for Topological Spaces
84. A Characterization of Normality
85. Separability Axioms
86. Second Countable Spaces
87. First Countable Spaces
88. Comparison of Topologies
89. Curysohn's Metrization Theorem

7. Compactness and Related Properties
90. Definitions of Various Compactness Properties
91. Some Consequences of Compactness
92. Relations Between Various Types of Compactness
93. Local Compactness
94. The One-Point Compactification
95. Some Generalizations of Mappings Defined on Compact Spaces

8. Connectedness and Related Concepts
96. Connectness. Definitions.
97. Some Basic Theorems Concerning Connectedness
98. Limit Superior and Limit Inferior of Sequences of Subsets of a Space
99. Review Questions

9. Quotient Spaces
100. Decomposition of a Topological Space
101. Quasi-Compact Mappings
102. The Quotient Topology
103. Decomposition of a Domain Space into Point Inverses
104. Topologically Equivalent Mappings
105. Decomposition of a Domain Space into Components of Point Inverses
106. Factorization of Compact Mappings

10. Net and Filter Convergence
107. Nets and Subnets
108. Convergence of Nets
109. Filters

11. Product Spaces
110. Cartesian Products
111. The Product Topology
112. Mappings into Product Spaces

References
Index

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## Source work progress

Exercises not all done