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The creative process of mathematics, both historically and individually, may be described as a counterpoint between theorems and examples. Al though it would be hazardous to claim that the creation of significant examples is less demanding than the development of theory, we have dis covered that focusing on examples is a particularly expeditious means of involving undergraduate mathematics students in actual research. Not only are examples more concrete than theorems-and thus more accessible-but they cut across individual theories and make it both appropriate and neces sary for the student to explore the entire literature in journals as well as texts. Indeed, much of the content of this book was first outlined by under graduate research teams working with the authors at Saint Olaf College during the summers of 1967 and 1968. In compiling and editing material for this book, both the authors and their undergraduate assistants realized a substantial increment in topologi cal insight as a direct result of chasing through details of each example. We hope our readers will have a similar experience. Each of the 143 examples in this book provides innumerable concrete illustrations of definitions, theo rems, and general methods of proof. There is no better way, for instance, to learn what the definition of metacompactness really means than to try to prove that Niemytzki's tangent disc topology is not metacompact. The search for counterexamples is as lively and creative an activity as can be found in mathematics research.
Contents
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 Ti Axioms.- Invariance Properties.- 4. Connectedness.- Functions and Products.- Disconnectedness.- Biconnectedness and Continua.- 5. Metric Spaces.- Complete Metric Spaces.- Metrizability.- Uniformities.- Metric Uniformities.- 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 [0,?) (? < ?).- 41. Closed Ordinal Space [0,?] (? < ?).- 42. Open Ordinal Space [0,?).- 43. Closed Ordinal Space [0,?].- 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 Order 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. Deleted 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. Deleted 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. ZZ.- 103. Uncountable Products of Z+.- 104. Baire Product Metric on Rw.- 105. II.- 106. [0,?) x II.- 107. Helly Space.- 108. C[0,1].- 109. Box Product Topology on Rw.- 110. Stone-?ech Compactification.- 111. Stone-?ech 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.- III Metrization Theory.- Conjectures and Counterexamples.- IV Appendices.- Special Reference Charts.- Separation Axiom Chart.- Compactness Chart.- Paracompactness Chart.- Connectedness Chart.- Disconnectedness Chart.- Metrizability Chart.- General Reference Chart.- Problems.- Notes.
