Counterexamples in Topology


Counterexamples in Topology is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr.
In the process of working on problems like the metrization problem, topologists have defined a wide variety of topological properties. It is often useful in the study and understanding of abstracts such as topological spaces to determine that one property does not follow from another. One of the easiest ways of doing this is to find a counterexample which exhibits one property but not the other. In Counterexamples in Topology, Steen and Seebach, together with five students in an undergraduate research project at St. Olaf College, Minnesota in the summer of 1967, canvassed the field of topology for such counterexamples and compiled them in an attempt to simplify the literature.
For instance, an example of a first-countable space which is not second-countable is counterexample #3, the discrete topology on an uncountable set. This particular counterexample shows that second-countability does not follow from first-countability.
Several other "Counterexamples in..." books and papers have followed, with similar motivations.

Reviews

In her review of the first edition, Mary Ellen Rudin wrote:
In his submission to Mathematical Reviews C. Wayne Patty wrote:
When the second edition appeared in 1978 its review in Advances in Mathematics treated topology as territory to be explored:

Notation

Several of the naming conventions in this book differ from more accepted modern conventions, particularly with respect to the separation axioms. The authors use the terms T3, T4, and T5 to refer to regular, normal, and completely normal. They also refer to completely Hausdorff as Urysohn. This was a result of the different historical development of metrization theory and general topology; see History of the separation axioms for more.
The Long Line in Example 45 is what most topologists nowerdays would call the Closed Long Ray.

List of mentioned 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. Sierpiński space, see also particular point topology
  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. Cantor set
  30. Rational numbers
  31. 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 where Γ<Ω
  41. Open ordinal space
  42. Uncountable discrete ordinal space
  43. Long line
  44. Extended long line
  45. An altered long line
  46. Lexicographic order topology on the unit square
  47. Right order topology
  48. Right order topology on R
  49. Right half-open interval topology
  50. Nested interval topology
  51. Overlapping interval topology
  52. Interlocking interval topology
  53. Hjalmar Ekdal topology, whose name was introduced in this book.
  54. Prime ideal topology
  55. Divisor topology
  56. Evenly spaced integer topology
  57. The p-adic topology on Z
  58. Relatively prime integer topology
  59. Prime integer topology
  60. Double pointed reals
  61. Countable complement extension topology
  62. Smirnov's deleted sequence topology
  63. Rational sequence topology
  64. Indiscrete rational extension of R
  65. Indiscrete irrational extension of R
  66. Pointed rational extension of R
  67. Pointed irrational extension of R
  68. Discrete rational extension of R
  69. Discrete irrational extension of R
  70. Rational extension in the plane
  71. Telophase topology
  72. Double origin topology
  73. Irrational slope topology
  74. Deleted diameter topology
  75. Deleted radius topology
  76. Half-disk topology
  77. Irregular lattice topology
  78. Arens square
  79. Simplified Arens square
  80. Niemytzki's tangent disk topology
  81. Metrizable tangent disk topology
  82. Sorgenfrey's half-open square topology
  83. Michael's product topology
  84. Tychonoff plank
  85. Deleted Tychonoff plank
  86. Alexandroff plank
  87. Dieudonné plank
  88. Tychonoff corkscrew
  89. Deleted Tychonoff corkscrew
  90. Hewitt's condensed corkscrew
  91. Thomas's plank
  92. Thomas's corkscrew
  93. Weak parallel line topology
  94. Strong parallel line topology
  95. Concentric circles
  96. Appert space
  97. Maximal compact topology
  98. Minimal Hausdorff topology
  99. Alexandroff square
  100. ZZ
  101. Uncountable products of Z+
  102. Baire product metric on Rω
  103. II
#
  1. Box product topology on Rω
  2. Stone–Čech compactification
  3. Stone–Čech compactification of the integers
  4. Novak space
  5. Strong ultrafilter topology
  6. Single ultrafilter topology
  7. Nested rectangles
  8. Topologist's sine curve
  9. Closed topologist's sine curve
  10. Extended topologist's sine curve
  11. Infinite broom
  12. Closed infinite broom
  13. Integer broom
  14. Nested angles
  15. Infinite cage
  16. Bernstein's connected sets
  17. Gustin's sequence space
  18. Roy's lattice space
  19. Roy's lattice subspace
  20. Cantor's leaky tent
  21. Cantor's teepee
  22. Pseudo-arc
  23. Miller's biconnected set
  24. Wheel without its hub
  25. Tangora's connected space
  26. Bounded metrics
  27. Sierpinski's metric space
  28. Duncan's space
  29. Cauchy completion
  30. Hausdorff's metric topology
  31. Post Office metric
  32. Radial metric
  33. Radial interval topology
  34. Bing's discrete extension space
  35. Michael's closed subspace