# 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.

### 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
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 $\left[{\,0, \Gamma}\right)$ $\paren {\Gamma < \Omega}$
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
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
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

Next

## 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 $4$

Every point is a component since every set containing more than one point is separated, ...

### 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