Book:Robert H. Kasriel/Undergraduate Topology

Subject Matter

 * Topology
 * Metric Spaces

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