# Book:Robert H. Kasriel/Undergraduate Topology

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## Robert H. Kasriel:

## Robert H. Kasriel: *Undergraduate Topology*

Published $1971$, **Dover Publications, Inc.**

- ISBN 0-486-47419-4.

### Subject Matter

### 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