Counterexamples in Topology

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,?) × 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.