Book:Lynn Arthur Steen/Counterexamples in Topology

Subject Matter

 * Topology
 * Metric Spaces

Contents

 * Preface
 * Preface to the Second Edition


 * PART I BASIC DEFINITIONS
 * 1. General Introduction
 * Limit Points
 * Closures and Interiors
 * Countability Properties
 * Functions
 * Filters
 * 2. Separation Axioms
 * Regular and Normal Spaces
 * Completely Hausdorff Spaces
 * Completely Regular Spaces
 * Functions, Products, and Subspaces
 * Additional Separation Properties
 * 3. Compactness
 * Global Compactness Properties
 * Localized Compactness Properties
 * Countability Axioms and Separability
 * Paracompactness
 * Compactness Properties and $T_i$ Axioms
 * Invariance Properties
 * 4. Connectedness
 * Functions and Products
 * Disconnectedness
 * Biconnectedness and Continua
 * 5. Metric Spaces
 * Complete Metric Spaces
 * Metrizability
 * Uniformities
 * Metric Uniformities


 * PART II: COUNTEREXAMPLES
 * 1. Finite Discrete Topology
 * 2. Countable Discrete Topology
 * 3. Uncountable Discrete Topology
 * 4. Indiscrete Topology
 * 5. Partition Topology
 * 6. Odd-Even Topology
 * 7. Deleted Integer Topology
 * 8. Finite Particular Point Topology
 * 9. Countable Particular Point Topology
 * 10. Uncountable Particular Point Topology
 * 11. Sierpinski Space
 * 12. Closed Extension Topology
 * 13. Finite Excluded Point Topology
 * 14. Countable Excluded Point Topology
 * 15. Uncountable Excluded Point Topology
 * 16. Open Extension Topology
 * 17. Either-Or Topology
 * 18. Finite Complement Topology on a Countable Space
 * 19. Finite Complement Topology on an Uncountable Space
 * 20. Countable Complement Topology
 * 21. Double Pointed Countable Complement Topology
 * 22. Compact Complement Topology
 * 23. Countable Fort Space
 * 24. Uncountable Fort Space
 * 25. Fortissimo Space
 * 26. Arens-Fort Space
 * 27. Modified Fort Space
 * 28. Euclidean Topology
 * 29. The Cantor Set
 * 30. The Rational Numbers
 * 31. The Irrational Numbers
 * 32. Special Subsets of the Real Line
 * 33. Special Subsets of the Plane
 * 34. One Point Compactification Topology
 * 35. One Point Compactification of the Rationals
 * 36. Hilbert Space
 * 37. Fréchet Space
 * 38. Hilbert Cube
 * 39. Order Topology
 * 40. Open Ordinal Space $\left[{\,0, \Gamma}\right)$ $\left({\Gamma < \Omega}\right)$
 * 41. Closed Ordinal Space $\left[{\,0, \Gamma}\right)$ $\left({\Gamma < \Omega}\right)$
 * 42. Open Ordinal Space $\left[{\,0, \Omega}\right)$
 * 43. Closed Ordinal Space $\left[{\,0, \Omega}\right)$
 * 44. Uncountable Discrete Ordinal Space
 * 45. The Long Line
 * 46. The Extended Long Line
 * 47. An Altered Long Line
 * 48. Lexicographic Ordering on the Unit Square
 * 49. Right Order Topology
 * 50. Right Older Topology on $R$
 * 51. Right Half-Open Interval Topology
 * 52. Nested Interval Topology
 * 53. Overlapping Interval Topology
 * 54. Interlocking Interval Topology
 * 55. Hjalmar Ekdal Topology
 * 56. Prime Ideal Topology
 * 57. Divisor Topology
 * 58. Evenly Spaced Integer Topology
 * 59. The $p$-adic Topology on $Z$
 * 60. Relatively Prime Integer Topology
 * 61. Prime Integer Topology
 * 62. Double Pointed Reals
 * 63. Countable Complement Extension Topology
 * 64. Smirnov's Deleted Sequence Topology
 * 65. Rational Sequence Topology
 * 66. Indiscrete Rational Extension of $R$
 * 67. Indiscrete Irrational Extension of $R$
 * 68. Pointed Rational Extension of $R$
 * 69. Pointed Irrational Extension of $R$
 * 70. Discrete Rational Extension of $R$
 * 71. Discrete Irrational Extension of $R$
 * 72. Rational Extension in the Plane
 * 73. Telophase Topology
 * 74. Double Origin Topology
 * 75. Irrational Slope Topology
 * 76. Delete Diameter Topology
 * 77. Deleted Radius Topology
 * 78. Half-Disc Topology
 * 79. Irregular Lattice Topology
 * 80. Arens Square
 * 81. Simplified Arens Square
 * 82. Niemytzki's Tangent Disc Topology
 * 83. Metrizable Tangent Disc Topology
 * 84. Sorgenfrey's Half-Open Square Topology
 * 85. Michael's Product Topology
 * 86. Tychonoff Plank
 * 87. Deleted Tychonoff Plank
 * 88. Alexandroff Plank
 * 89. Dieudonne Plank
 * 90. Tychonoff Corkscrew
 * 91. Delete Tychonoff Corkscrew
 * 92. Hewitt's Condensed Corkscrew
 * 93. Thomas' Plank
 * 94. Thomas' Corkscrew
 * 95. Weak Parallel Line Topology
 * 96. Strong Parallel Line Topology
 * 97. Concentric Circles
 * 98. Appert Space
 * 99. Maximal Compact Topology
 * 100. Minimal Hausdorff Topology
 * 101. Alexandroff Square
 * 102. $Z^Z$
 * 103. Uncountable Products of $Z^+$
 * 104. Baire Product Metric on $R^\omega$
 * 105. $I^I$
 * 106. $\left[{\,0, \Omega}\right) \times I^I$
 * 107. Helly Space
 * 108. $C \left[{0, 1}\right]$
 * 109. Box Product Topology on $R^\omega$
 * 110. Stone-Čech Compactification
 * 111. Stone-Čech Compactification of the Integers
 * 112. Novak Space
 * 113. Strong Ultrafilter Topology
 * 114. Single Ultrafilter Topology
 * 115. Nested Rectangles
 * 116. Topologist's Sine Curve
 * 117. Closed Topologist's Sine Curve
 * 118. Extended Topologist's Sine Curve
 * 119. The Infinite Broom
 * 120. The Closed Infinite Broom
 * 121. The Integer Broom
 * 122. Nested Angles
 * 123. The Infinite Cage
 * 124. Bernstein's Connected Sets
 * 125. Gustin's Sequence Space
 * 126. Roy's Lattice Space
 * 127. Roy's Lattice Subspace
 * 128. Cantor's Leaky Tent
 * 129. Cantor's Teepee
 * 130. A Pseudo-Arc
 * 131. Miller's Biconnected Set
 * 132. Wheel without Its Hub
 * 133. Tangora's Connected Space
 * 134. Bounded Metrics
 * 135. Sierpinski's Metric Space
 * 136. Duncan's Space
 * 137. Cauchy Completion
 * 138. Hausdorff's Metric Topology
 * 139. The Post Office Metric
 * 140. The Radial Metric
 * 141. Radial Interval Topology
 * 142. Bing's Discrete Extension Space
 * 143. Michael's Closed Subspace


 * PART III: METRIZATION THEORY
 * Conjectures and Counterexamples


 * PART IV: APPENDICES
 * Special Reference Charts
 * Separation Axiom Chart
 * Compactness Chart
 * Paracompactness Chart
 * Connectedness Chart
 * Disconnectedness Chart
 * Metrizability Chart
 * General Reference Chart
 * Problems
 * Notes
 * Bibliography


 * Index

Known Mistakes

 * Part $\text{I}: \ \S 1$: Closures and Interiors: Union of Exteriors contains Exterior of Intersection


 * Part $\text{I}: \ \S 3$: Invariance Properties: Compactness Properties Preserved under Continuous Mappings


 * Part $\text{I}: \ \S 4$: A topological space is defined as being locally connected if it has a basis consisting entirely of connected sets.
 * The true definition is that each point has a local basis consisting entirely of connected sets.


 * Equivalence of metrics is not defined, although the concept is mentioned and used in the context of complete metric spaces in part $\text{I}: \ \S 5$: Complete Metric Spaces.


 * Part $\text{I}: \ \S 5$: Uniformities: The definition of a uniformity contains an incorrect statement. It defines the inverse $u^{-1}$ of an entourage $u$ of a uniformity $\mathcal U$ as:


 * $u^{-1} := \left\{{\left({y, x}\right): \left({x, y}\right) \in \mathcal U}\right\}$
 * ... whereas it ought to be:


 * $u^{-1} := \left\{{\left({y, x}\right): \left({x, y}\right) \in u}\right\}$


 * Part $\text{II}: \ \S 8 - 10: \ 4$: Non-Trivial Particular Point Topology is not T4
 * The particular point space with two points, that is, the Sierpiński space, satisfies the $T_4$ axiom. The statement is true for a particular point space with three or more points.


 * Part $\text{II}: \ \S 8 - 10: \ 12$: Particular Point Space is not Weakly Countably Compact
 * This statement applies only to an infinite particular point space. The finite case, through being finite, satisfies all compactness properties.


 * Part $\text{II}: \ \S 27: \ 4$: Sets in Modified Fort Space are Separated‎
 * The correct term is disconnected, not separated.


 * Part $\text{II}: \ \S 32: \ 9$: Kuratowski's Closure-Complement Problem
 * The graphic presented as Figure $12$ is incorrect in places.