ProofWiki:Books/Lynn Arthur Steen/Counterexamples in Topology

Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology

Published 1970, 2nd Edition 1978, Dover Publications, Inc..

ISBN 0-486-68735-X

Subject Matter

• Topology
• Metric Spaces

Contents

Preface

Preface to the Second Edition

PART 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 $T_i$ Axioms

Invariance Properties

4. Connectedness

Functions and Products

Disconnectedness

Biconnectedness and Continua

5. Metric Spaces

Complete Metric Spaces

Metrizability

Uniformities

Metric Uniformities

PART 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 $\left[{\,0, \Gamma}\right)$ $\left({\Gamma < \Omega}\right)$

41. Closed Ordinal Space $\left[{\,0, \Gamma}\right)$ $\left({\Gamma < \Omega}\right)$

42. Open Ordinal Space $\left[{\,0, \Omega}\right)$

43. Closed Ordinal Space $\left[{\,0, \Omega}\right)$

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 Older 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. Delete 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. Delete 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. $Z^Z$

103. Uncountable Products of $Z^+$

104. Baire Product Metric on $R^\omega$

105. $I^I$

106. $\left[{\,0, \Omega}\right) \times I^I$

107. Helly Space

108. $C \left[{0, 1}\right]$

109. Box Product Topology on $R^\omega$

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. 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

PART III: METRIZATION THEORY

Conjectures and Counterexamples

PART IV: APPENDICES

Special Reference Charts

Separation Axiom Chart

Compactness Chart

Paracompactness Chart

Connectedness Chart

Disconnectedness Chart

Metrizability Chart

General Reference Chart

Problems

Notes

Bibliography

Index

Known Errors

• Part $\text{I}: \ \S 1$: Closures and Interiors: Union of Exteriors contains Exterior of Intersection

• Part $\text{I}: \ \S 3$: Invariance Properties: Compactness Properties Preserved under Continuous Mappings

• Part $\text{I}: \ \S 4$: A topological space is defined as being locally connected if it has a basis consisting entirely of connected sets.

The true definition is that each point has a local basis consisting entirely of connected sets.

• Equivalence of metrics is not defined, although the concept is mentioned and used in the context of complete metric spaces in part $\text{I}: \ \S 5$: Complete Metric Spaces.

• Part $\text{I}: \ \S 5$: Uniformities: The definition of a uniformity contains an incorrect statement. It defines the inverse $u^{-1}$ of an entourage $u$ of a uniformity $\mathcal U$ as:

$u^{-1} := \left\{{\left({y, x}\right): \left({x, y}\right) \in \mathcal U}\right\}$

... whereas it ought to be:

$u^{-1} := \left\{{\left({y, x}\right): \left({x, y}\right) \in u}\right\}$

• Part $\text{II}: \ \S 8 - 10: \ 4$: Non-Trivial Particular Point Topology is not T4

The particular point space with two points, that is, the Sierpiński space, satisfies the $T_4$ axiom. The statement is true for a particular point space with three or more points.

• Part $\text{II}: \ \S 8 - 10: \ 12$: Particular Point Space is Not Weakly Countably Compact

This statement applies only to an infinite particular point space. The finite case, through being finite, satisfies all compactness properties.

• Part $\text{II}: \ \S 27: \ 4$: Sets in Modified Fort Space are Separated‎

The correct term is disconnected, not separated.