# Book:Lynn Arthur Steen/Counterexamples in Topology/Second Edition

## Lynn Arthur Steen and J. Arthur Seebach, Jr.: *Counterexamples in Topology (2nd Edition)*

Published $\text {1978}$, **Dover Publications, Inc.**

- ISBN 0-486-68735-X.

### Subject Matter

### Contents

- Preface
- Preface to the Second Edition

**PART I BASIC DEFINITIONS**

**1. General Introduction**

**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

**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 $\left[{\,0, \Gamma}\right)$ $\paren {\Gamma < \Omega}$
- 41. Closed Ordinal Space $\left[{\,0, \Gamma}\right]$ $\paren {\Gamma < \Omega}$
- 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 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. Dieudonné 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. $Z^Z$
- 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*

## Further Editions

## Errata

### Union of Exteriors contains Exterior of Intersection

Part $\text I$: Basic Definitions: Section $1.$ General Introduction: Closures and Interiors

*The exterior of the union of sets is always contained in the intersection of the exteriors, and similarly, the exterior of the intersection is contained in the union of the exteriors; equality holds only for finite unions and intersections.*

### Compactness Properties Preserved under Continuous Mapping

Part $\text I$: Basic Definitions: Section $3$. Compactness: Invariance Properties

*To be precise, the properties of compactness, $\sigma$-compactness, countable compactness, sequential compactness, Lindelöf, and separability are preserved under continuous maps ... [Weak] local compactness, and first and second countability are preserved under open continuous maps, but not just under continuous maps ...*

### Equivalence of Metrics is not Defined

Part $\text I$: Basic Definitions: Section $5$. Metric Spaces: Complete Metric Spaces

- The concept of
**equivalence of metrics**is not defined, although the concept is mentioned and used in the context of complete metric spaces.

### Definition of Uniformity

Part $\text I$: Basic Definitions: Section $5$. Metric Spaces: Uniformities

*The quasiuniformity $\UU$ is a***uniformity**if the following additional condition is satisfied:*$\text U 5$: If $u \in \UU$, then $u^{-1} \in \UU$ where $u^{-1} = \set {\tuple {y, x}: \tuple {x, y} \in \UU}$.*

### Accumulation Points of Sequence of Distinct Terms in Infinite Particular Point Space

Part $\text {II}$: Counterexamples: $9 \text { - } 10$: Infinite Particular Point Topology: Item $1$

*The sequences $\set {a_i}$ which converge are those for which the $a_i \ne p$ are equal for all but a finite number of indices. The only accumulation points for sequences are the points $b_j$ that the $a_i$ equal for infinitely many indices.*

### Non-Trivial Particular Point Topology is not $T_4$

Part $\text {II}$: Counterexamples: $8 \text { - } 10$: Particular Point Topology: Item $4$

*Every particular point topology is $T_0$, but since there are no disjoint open sets, none of the higher separation axioms are satisfied unless $X$ has only one point.*

### Particular Point Space is not Weakly Countably Compact

Part $\text {II}$: Counterexamples: $8 \text { - } 10$: Particular Point Topology: Item $12$

*[A particular point space] $X$ is not weakly countably compact since any set which does not contain $p$ has no limit points.*

### Sets in Modified Fort Space are Separated

Part $\text{II}$: Counterexamples: $27$: Modified Fort Space: Item $4$

### Kuratowski's Closure-Complement Problem

Part $\text {II}$: Counterexamples: $32$: Special Subsets of the Real Line: Item $9$: Figure $12$

Steen and Seebach present a more complicated $14$-set than is necessary to demonstrate the theorem:

- $A := \set {\tfrac 1 n: n \in \Z_{>0} } \cup \openint 2 3 \cup \openint 3 4 \cup \set {4 \tfrac 1 2} \cup \closedint 5 6 \cup \paren {\hointr 7 8 \cap \Q}$

They present Figure $12$ to illustrate the various generated subsets graphically:

The following mistakes can be identified in the above diagram:

- $(1): \quad$ The set $A$ as presented expresses the interval of rationals as closed, whereas it is in fact half open.

- $(2): \quad$ The sets are all presented as subsets of $\R_{\ge 0}$, while this is not stated in the text.

- $(3): \quad$ $0$ is erroneously excluded from $A^{\prime}$.

### Complement of Set of Rational Pairs in Real Euclidean Plane is Arc-Connected

Part $\text{II}$: Counterexamples: $33$: Special Subsets of the Plane: Item $2$

*Let $A$ be the subset of $\R^2 = \R \times \R$ consisting of all point with at least one irrational coordinate, and let $A$ have the induced topology. $A$ is arc-connected since a point $\tuple {x_1, y_1}$ with two irrational coordinates may be joined by an arc to any point $\tuple {a, b} \in A$ either $a$ or $b$ is irrational, say $a$. Then the union of the lines $x = a, y = y_1$ is an arc-connected subset of $X$ connecting $\tuple {x_1, y_1}$ to $\tuple {a, b}$. ...*

### Linearly Ordered Space is $T_5$

Part $\text {II}$: Counterexamples: $39$. Order Topology: Item $6$

*For each $\gamma$, select and fix some point $k_\gamma \in C_\gamma$. Then whenever $A_\alpha \cap \overline S_\alpha \ne \O$, there exists a unique ${k_\alpha}^+ \in {C_\alpha}^+$, the immediate successor of $A_\alpha$ ... otherwise, if $A_\alpha \cap {\overline S_\alpha}^\alpha = \O$, let $I_\alpha = \O$. ...*

### Telophase Topology

Part $\text {II}$: Counterexamples: $73$. Telophase Topology

*Let $\struct {X, \tau}$ be the topological space formed by adding to the ordinary closed unit topology $\sqbrk {0, 1}$ another right end point, say $1^*$, with the sets $\paren {\alpha, 1} \cup \set {1^*}$ as a local neighborhood basis.*

### Bibliography: Appert

Bibliography

- [10] Appert, Q.
*Propriétés des Espaces Abstraits les Plus Généraux*. Actual. Sci. Ind. No. 146, Herman, 1934

## Source work progress

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $42$. Open Ordinal Space $[0, \Omega)$: $10$