Counterexamples in Topology

Counterexamples in Topology (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr.

Counterexamples in Topology
Counterexamples in Topology.jpg
AuthorLynn Arthur Steen
J. Arthur Seebach, Jr.
CountryUnited States
SubjectTopological spaces
Publication date
Media typeHardback, Paperback
Pages244 pp.
514/.3 20
LC ClassQA611.3 .S74 1995

In the process of working on problems like the metrization problem, topologists (including Steen and Seebach) 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.


In her review of the first edition, Mary Ellen Rudin wrote:

In other mathematical fields one restricts one's problem by requiring that the space be Hausdorff or paracompact or metric, and usually one doesn't really care which, so long as the restriction is strong enough to avoid this dense forest of counterexamples. A usable map of the forest is a fine thing...[1]

In his submission[2] to Mathematical Reviews C. Wayne Patty wrote:

...the book is extremely useful, and the general topology student will no doubt find it very valuable. In addition it is very well written.

When the second edition appeared in 1978 its review in Advances in Mathematics treated topology as territory to be explored:

Lebesgue once said that every mathematician should be something of a naturalist. This book, the updated journal of a continuing expedition to the never-never land of general topology, should appeal to the latent naturalist in every mathematician.[3]


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 nowadays would call the 'closed long ray'.

List of mentioned counterexamplesEdit

  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 [0,Γ) where Γ<Ω
  41. Closed ordinal space [0,Γ] where Γ<Ω
  42. Open ordinal space [0,Ω)
  43. Closed ordinal space [0,Ω]
  44. Uncountable discrete ordinal space
  45. Long line
  46. Extended long line
  47. An altered long line
  48. Lexicographic order topology 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, whose name was introduced in this book.
  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-disk topology
  79. Irregular lattice topology
  80. Arens square
  81. Simplified Arens square
  82. Niemytzki's tangent disk topology
  83. Metrizable tangent disk topology
  84. Sorgenfrey's half-open square topology
  85. Michael's product topology
  86. Tychonoff plank
  87. Deleted Tychonoff plank
  88. Alexandroff plank
  89. Dieudonné plank
  90. Tychonoff corkscrew
  91. Deleted Tychonoff corkscrew
  92. Hewitt's condensed corkscrew
  93. Thomas's plank
  94. Thomas's 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 Rω
  105. II
  106. [0,Ω)×II
  107. Helly space
  108. C[0,1]
  109. Box product topology on Rω
  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. Infinite broom
  120. Closed infinite broom
  121. Integer broom
  122. Nested angles
  123. 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. 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. Post Office metric
  140. Radial metric
  141. Radial interval topology
  142. Bing's discrete extension space
  143. Michael's closed subspace

See alsoEdit


  1. ^ Rudin, Mary Ellen (1971). "Review: Counterexamples in Topology". American Mathematical Monthly. 78 (7). pp. 803–804. doi:10.2307/2318037. MR 1536430.
  2. ^ C. Wayne Patty (1971) "Review: Counterexamples in Topology", MR0266131
  3. ^ Kung, Joseph; Rota, Gian-Carlo (1979). "Review: Counterexamples in Topology". Advances in Mathematics. 32 (1). p. 81. doi:10.1016/0001-8708(79)90031-8.
  • Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).

External linksEdit