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Counterexamples in Topology

From Wikipedia, the free encyclopedia
Book by Lynn Steen
Counterexamples in Topology
AuthorLynn Arthur Steen
J. Arthur Seebach, Jr.
LanguageEnglish
SubjectTopological spaces
GenreNon-fiction
PublisherSpringer-Verlag
Publication date
1970
Publication placeUnited States
Media typeHardback,Paperback
Pages244 pp.
ISBN0-486-68735-X
OCLC32311847
514/.3 20
LC ClassQA611.3 .S74 1995

Counterexamples in Topology (1970, 2nd ed. 1978) is a book onmathematics bytopologistsLynn Steen andJ. Arthur Seebach, Jr.

In the process of working on problems like themetrization problem, topologists (including Steen and Seebach) have defined a wide variety oftopological properties. It is often useful in the study and understanding of abstracts such astopological spaces to determine that one property does not follow from another. One of the easiest ways of doing this is to find acounterexample which exhibits one property but not the other. InCounterexamples in Topology, Steen and Seebach, together with five students in an undergraduate research project atSt. Olaf College,Minnesota in the summer of 1967, canvassed the field oftopology for such counterexamples and compiled them in an attempt to simplify the literature.

For instance, an example of afirst-countable space which is notsecond-countable is counterexample #3, thediscrete topology on anuncountable 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

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

In other mathematical fields one restricts one's problem by requiring that thespace beHausdorff orparacompact ormetric, 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] toMathematical 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 inAdvances in Mathematics treated topology as territory to be explored:

Lebesgue once said that every mathematician should be something of anaturalist. 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]

Notation

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Several of thenaming conventions in this book differ from more accepted modern conventions, particularly with respect to theseparation axioms. The authors use the terms T3, T4, and T5 to refer toregular,normal, andcompletely normal. They also refer tocompletely Hausdorff asUrysohn. This was a result of the different historical development of metrization theory andgeneral topology; seeHistory of the separation axioms for more.

Thelong line in example 45 is what most topologists nowadays would call the 'closed long ray'.

List of mentioned counterexamples

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  1. Finitediscrete topology
  2. Countablediscrete topology
  3. Uncountablediscrete 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 alsoparticular point topology
  12. Closed extension topology
  13. Finiteexcluded point topology
  14. Countableexcluded point topology
  15. Uncountableexcluded point topology
  16. Open extension topology
  17. Either-or topology
  18. Finite complement topology on acountable space
  19. Finite complement topology on an uncountable space
  20. Countable complement topology
  21. Double pointedcountable complement topology
  22. Compact complement topology
  23. CountableFort space
  24. UncountableFort space
  25. Fortissimo space
  26. Arens–Fort space
  27. ModifiedFort 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 alteredlong line
  48. Lexicographic order topology on the unit square
  49. Right order topology
  50. Right order topology onR
  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. Thep-adic topology onZ
  60. Relativelyprime 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 ofR
  67. Indiscrete irrational extension ofR
  68. Pointed rational extension ofR
  69. Pointed irrational extension ofR
  70. Discrete rational extension ofR
  71. Discrete irrational extension ofR
  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. SimplifiedArens 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. MinimalHausdorff topology
  101. Alexandroff square
  102. ZZ
  103. Uncountable products ofZ+
  104. Baire product metric onRω
  105. II
  106. [0,Ω)×II
  107. Helly space
  108. C[0,1]
  109. Box product topology onRω
  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 also

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References

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  1. ^Rudin, Mary Ellen (1971). "Review:Counterexamples in Topology".American Mathematical Monthly.78 (7):803–804.doi:10.2307/2318037.JSTOR 2318037.MR 1536430.
  2. ^C. Wayne Patty (1971) "Review:Counterexamples in Topology",MR 0266131
  3. ^Kung, Joseph;Rota, Gian-Carlo (1979)."Review:Counterexamples in Topology".Advances in Mathematics.32 (1): 81.doi:10.1016/0001-8708(79)90031-8.

Bibliography

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External links

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