Counterexamples in Topology, Second Edition

Over 140 examples, preceded by a succinct exposition of general topology and basic terminology. Each example treated as a whole. Over 25 Venn diagrams and charts summarize properties of the examples, while discussions of general methods of construction and change give readers insight into constructing counterexamples. Extensive collection of problems and exercises, correlated with examples. Bibliography. 1978 edition. Cover......Page 1 Counterexamples in Topology......Page 4 ISBN 048668735X......Page 5 Preface......Page 6 Contents......Page 10 Part I BASIC DEFINITIONS......Page 16 1. General Introduction......Page 18 Limit Points......Page 20 Closures and Interiors......Page 21 Functions......Page 22 Filters......Page 24 2. Separation Axioms......Page 26 Regular and Normal Spaces......Page 27 Completely Regular Spaces......Page 28 Functions, Products, and Subspaces......Page 29 Additional Separation Properties......Page 31 Global Compactness Properties......Page 33 Localized Compactness Properties......Page 35 Countability Axioms and Separability......Page 36 Paracompactness......Page 37 Compactness Properties and T-sub-i Axioms......Page 39 Invariance Properties......Page 41 4. Connectedness......Page 43 Disconnectedess......Page 46 Biconnectedness and Continua......Page 48 5. Metric Spaces......Page 49 Complete Metric Spaces......Page 51 Uniformities......Page 52 Metric Uniformities......Page 53 Part II COUNTEREXAMPLES......Page 54 3. Uncountable Discrete Topology......Page 56 4. Indiscrete Topology......Page 57 7. Deleted Integer Topology......Page 58 12. Closed Extension Topology......Page 59 16. Open Extension Topology......Page 62 17. Either-Or Topology......Page 63 19. Finite Complement Topology on an Uncountable Space......Page 64 21. Double Pointed Countable Complement Topology......Page 65 22. Compact Complement Topology......Page 66 24. Uncountable Fort Space......Page 67 25. Fortissimo Space......Page 68 26. Arens-Fort Space......Page 69 27. Modified Fort Space......Page 70 28. Euclidean Topology......Page 71 29. The Cantor Set......Page 72 31. The Irrational Numbers......Page 74 32. Special Subsets of the Real Line......Page 75 33. Special Subsets of the Plane......Page 76 35. One Point Compactification of the Rationals......Page 78 37. Frechet Space......Page 79 38. Hilbert Cube......Page 80 39. Order Topology......Page 81 43. Closed Ordinal Space [0,0]......Page 83 44. Uncountable Discrete Ordinal Space......Page 85 46. The Extended Long Line......Page 86 47. An Altered Long Line......Page 87 48. Lexicographic Ordering on the Unit Square......Page 88 50. Right Order Topology on R......Page 89 51. Right Half-Open Interval Topology......Page 90 52. Nested Interval Topology......Page 91 54. Interlocking Interval Topology......Page 92 55. Hjalmar Ekdal Topology......Page 93 57. Divisor Topology......Page 94 58. Evenly Spaced Integer Topology......Page 95 59. The p-adic Topology on Z......Page 96 61. Prime Integer Topology......Page 97 62. Double Pointed Reals......Page 99 63. Countable Complement Extension Topology......Page 100 64. Smirnov's Deleted Sequence Topology......Page 101 65. Rational Sequence Topology......Page 102 69. Pointed Irrational Extension of R......Page 103 71. Discrete Irrational Extension of R......Page 105 72. Rational Extension in the Plane......Page 106 74. Double Origin Topology......Page 107 75. Irrational Slope Topology......Page 108 77. Deleted Radius Topology......Page 109 78. Half-Disc Topology......Page 111 79. Irregular Lattice Topology......Page 112 80. Arens Square......Page 113 82. Niemytzki's Tangent Disc Topology......Page 115 84. Sorgenfrey's Half-Open Square Topology......Page 118 85. Michael's Product Topology......Page 120 87. Deleted Tychonoff Plank......Page 121 88. Alexandroff Plank......Page 122 89. Dieudonne Plank......Page 123 91. Deleted Tychonoff Corkscrew......Page 124 92. Hewitt's Condensed Corkscrew......Page 126 94. Thomas' Corkscrew......Page 128 96. Strong Parallel Line Topology......Page 129 97. Concentric Circles......Page 131 98. Appert Space......Page 132 99. Maximal Compact Topology......Page 133 100. Minimal Hausdorff Topology......Page 134 101. Alexandroff Square......Page 135 102. Z^Z......Page 136 103. Uncountable Products of Z+......Page 138 104. Baire Product Metric on Rw......Page 139 105. I^I......Page 140 106. [O,Omega) X I^I......Page 141 107. Helly Space......Page 142 109. Box Product Topology on Rw......Page 143 110. Stone-Cech Compactification......Page 144 111. Stone-Cech Compactification of the Integers......Page 147 112. Novak Space......Page 149 113. Strong Ultrafilter Topology......Page 150 114. Single Ultrafilter Topology......Page 151 118. Extended Topologist's Sine Curve......Page 152 120. The Closed Infinite Broom......Page 154 122. Nested Angles......Page 155 123. The Infinite Cage......Page 156 125. Gustin's Sequence Space......Page 157 127. Roy's Lattice Subspace......Page 158 129. Cantor's Teepee......Page 160 130. A Pseudo-Arc......Page 162 131. Miller's Biconnected Set......Page 163 133. Tangora's Connected Space......Page 165 134. Bounded Metrics......Page 166 135. Sierpinski's Metric Space......Page 167 136. Duncan's Space......Page 168 138. Hausdorff's Metric Topology......Page 169 140. The Radial Metric......Page 170 141. Radial Interval Topology......Page 171 143. Michael's Closed Subspace......Page 172 Part III METRIZATION THEORY......Page 174 Conjectures and Counterexamples......Page 176 Part IV APPENDICES......Page 198 Special Reference Charts......Page 200 Separation Axiom Chart......Page 202 Compactness Chart......Page 203 Paracompactness Chart......Page 205 Connectedness Chart......Page 206 Disconnectedness Chart......Page 207 Metrizability Chart......Page 208 General Reference Chart......Page 210 Problems......Page 220 Notes......Page 228 Bibliography......Page 243 Index......Page 251 According to the authors of this highly useful compendium, focusing on examples is an extremely effective method of involving undergraduate mathematics students in actual research. It is only as a result of pursuing the details of each example that students experience a significant increment in topological understanding. With that in mind, Professors Steen and Seebach have assembled 143 examples in this book, providing innumerable concrete illustrations of definitions, theorems, and general methods of proof. Far from presenting all relevant examples, however, the book instead provides a fruitful context in which to ask new questions and seek new answers.Ranging from the familiar to the obscure, the examples are preceded by a succinct exposition of general topology and basic terminology and theory. Each example is treated as a whole, with a highly geometric exposition that helps readers comprehend the material. Over 25 Venn diagrams and reference charts summarize the properties of the examples and allow students to scan quickly for examples with prescribed properties. In addition, discussions of general methods of constructing and changing examples acquaint readers with the art of constructing counterexamples. The authors have included an extensive collection of problems and exercises, all correlated with various examples, and a bibliography of 140 sources, tracing each uncommon example to its origin.This revised and expanded second edition will be especially useful as a course supplement and reference work for students of general topology. Moreover, it gives the instructor the flexibility to design his own course while providing students with a wealth of historically and mathematically significant examples. 1978 edition. According to the authors of this highly useful compendium, focusing on examples is an extremely effective method of involving undergraduate mathematics students in actual research. It is only as a result of pursuing the details of each example that students experience a significant increment in topological understanding. With that in mind, Professors Steen and Seebach have assembled 143 examples in this book, providing innumerable concrete illustrations of definitions, theorems, and general methods of proof. Far from presenting all relevant examples, however, the book instead provides a fruitful context in which to ask new questions and seek new answers. Ranging from the familiar to the obscure, the examples are preceded by a succinct exposition of general topology and basic terminology and theory. Each example is treated as a whole, with a highly geometric exposition that helps readers comprehend the material. Over 25 Venn diagrams and reference charts summarize the properties of the examples and allow students to scan quickly for examples with prescribed properties. In addition, discussions of general methods of constructing and changing examples acquaint readers with the art of constructing counterexamples. The authors have included an extensive collection of problems and exercises, all correlated with various examples, and a bibliography of 140 sources, tracing each uncommon example to its origin. This revised and expanded second edition will be especially useful as a course supplement and reference work for students of general topology. Moreover, it gives the instructor the flexibility to design his own course while providing students with a wealth of historically and mathematically significant examples Over 140 examples, preceded by a succinct exposition of general topology and basic terminology. Each example treated as a whole. Over 25 Venn diagrams and charts summarize properties of the examples, while discussions of general methods of construction and change give readers insight into constructing counterexamples. Includes problems and exercises, correlated with examples. Bibliography. 1978 edition. "Over 140 examples, preceded by a succinct exposition of general topology and basic terminology. Each example treated as a whole. Over 25 Venn diagrams and charts summarize properties of the examples, while discussions of general methods of construction and change give readers insight into constructing counterexamples."--Pub. desc