Book:Lynn Arthur Steen/Counterexamples in Topology/Second Edition

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



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

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

Space is Locally Connected iff it has Basis of Connected Sets
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
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.

Definition of Uniformity
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 $\UU$ as:


 * $u^{-1} := \set {\tuple {y, x}: \tuple {x, y} \in \UU}$

... whereas it ought to be:


 * $u^{-1} := \set {\tuple {y, x}: \tuple {x, y} \in u}$

Non-Trivial Particular Point Topology is not T4
Part $\text{II}: \ \S 8 - 10: \ 4$:

Particular Point Space is not Weakly Countably Compact
Part $\text{II}: \ \S 8 - 10: \ 12$:

Sets in Modified Fort Space are Separated‎
Part $\text{II}: \ \S 27: \ 4$:

Kuratowski's Closure-Complement Problem‎
Part $\text{II}: \ \S 32: \ 9$:

Complement of Set of Rational Pairs in Real Euclidean Plane is Arc-Connected‎
Part $\text{II}: \ \S 33: \ 2$:

Source work progress
* : $\text{II}: \ 41: \ 7$


 * Need to go through and correct the edition throughout -- it has been cited as the $1970$ consistently So far:


 * : Part $\text I$: Basic Definitions: Section $1$: General Introduction: Closures and Interiors