ProofWiki:Books/Lynn Arthur Steen/Counterexamples in Topology

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Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology

Published 1970, 2nd Edition 1978, Dover Publications, Inc..

ISBN 0-486-68735-X


Subject Matter


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 Errors

The true definition is that each point has a local basis consisting entirely of connected sets.
$u^{-1} := \left\{{\left({y, x}\right): \left({x, y}\right) \in \mathcal U}\right\}$
... whereas it ought to be:
$u^{-1} := \left\{{\left({y, x}\right): \left({x, y}\right) \in u}\right\}$
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.
This statement applies only to an infinite particular point space. The finite case, through being finite, satisfies all compactness properties.
The correct term is disconnected, not separated.