Counterexamples in Topology (1970, 2nd ed. 1978) is a book on mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 by topologist

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

s Lynn Steen

Lynn Steen

Lynn Arthur Steen is an American mathematician who is Professor of Mathematics at St. Olaf College, Northfield, Minnesota in the U.S. He has written numerous books and articles on the teaching of mathematics...

 and J. Arthur Seebach, Jr.

J. Arthur Seebach, Jr.

J. Arthur Seebach, Jr was an American mathematician.He received his Ph.D. in 1968 from Northwestern University. He joined the faculty of Mathematics at St. Olaf College in 1965. He was Associate Editor of the American Mathematical Monthly from 1971 to 1986 and Editor of Mathematics Magazine from...

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 space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

s to determine that one property does not follow from another. One of the easiest ways of doing this is to find a counterexample

Counterexample

In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule. For example, consider the proposition "all students are lazy"....

 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

St. Olaf College

St. Olaf College is a coeducational, residential, four-year, private liberal arts college in Northfield, Minnesota, United States. It was founded in 1874 by a group of Norwegian-American immigrant pastors and farmers, led by Pastor Bernt Julius Muus. The college is named after Olaf II of Norway,...

, Minnesota

Minnesota

Minnesota is a U.S. state located in the Midwestern United States. The twelfth largest state of the U.S., it is the twenty-first most populous, with 5.3 million residents. Minnesota was carved out of the eastern half of the Minnesota Territory and admitted to the Union as the thirty-second state...

 in the summer of 1967, canvassed the field of topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

 for such counterexamples and compiled them in an attempt to simplify the literature.

For instance, an example of a first-countable space

First-countable space

In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis...

 which is not second-countable

Second-countable space

In topology, a second-countable space, also called a completely separable space, is a topological space satisfying the second axiom of countability. A space is said to be second-countable if its topology has a countable base...

 is counterexample #3, the discrete topology on an uncountable set

Uncountable set

In mathematics, an uncountable set is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.-Characterizations:There...

. 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.

Notation

Several of the naming convention

Naming convention

A naming convention is a convention for naming things. The intent is to allow useful information to be deduced from the names based on regularities. For instance, in Manhattan, streets are numbered, with East-West streets being called "Streets" and North-South streets called "Avenues".-Use...

s in this book differ from more accepted modern conventions, particularly with respect to the separation axiom

Separation axiom

In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms...

s. The authors use the terms T₃, T₄, and T₅ to refer to regular

Regular space

In topology and related fields of mathematics, a topological space X is called a regular space if every non-empty closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods. Thus p and C...

, normal

Normal space

In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space...

, and completely normal. They also refer to completely Hausdorff

Completely Hausdorff space

In topology, an Urysohn space, or T2½ space, is a topological space in which any two distinct points can be separated by closed neighborhoods. A completely Hausdorff space, or functionally Hausdorff space, is a topological space in which any two distinct points can be separated by a continuous...

 as Urysohn. This was a result of the different historical development of metrization theory and general topology

General topology

In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...

; see History of the separation axioms

History of the separation axioms

In general topology, the separation axioms have had a convoluted history, with many competing meanings for the same term, and many competing terms for the same concept.- Origins :...

 for more.

List of mentioned counterexamples

1. Finite discrete topology
2. Countable discrete topology
3. Uncountable discrete topology
4. Indiscrete topology
5. Partition topology
   Partition topology
   In mathematics, the partition topology is a topology that can be induced on any set X by partitioning X into disjoint subsets P; these subsets form the basis for the topology...
   
   
6. Odd–even topology
7. Deleted integer topology
8. Finite particular point topology
   Particular point topology
   In mathematics, the particular point topology is a topology where sets are considered open if they are empty or contain a particular, arbitrarily chosen, point of the topological space. Formally, let X be any set and p ∈ X. The collectionof subsets of X is then the particular point topology...
   
   
9. Countable particular point topology
   Particular point topology
   In mathematics, the particular point topology is a topology where sets are considered open if they are empty or contain a particular, arbitrarily chosen, point of the topological space. Formally, let X be any set and p ∈ X. The collectionof subsets of X is then the particular point topology...
   
   
10. Uncountable particular point topology
    Particular point topology
    In mathematics, the particular point topology is a topology where sets are considered open if they are empty or contain a particular, arbitrarily chosen, point of the topological space. Formally, let X be any set and p ∈ X. The collectionof subsets of X is then the particular point topology...
    
    
11. Sierpinski space
    Sierpinski space
    In mathematics, the Sierpiński space is a finite topological space with two points, only one of which is closed.It is the smallest example of a topological space which is neither trivial nor discrete...
    
    , see also particular point topology
    Particular point topology
    In mathematics, the particular point topology is a topology where sets are considered open if they are empty or contain a particular, arbitrarily chosen, point of the topological space. Formally, let X be any set and p ∈ X. The collectionof subsets of X is then the particular point topology...
    
    
12. Closed extension topology
    Closed extension topology
    In topology, a branch of mathematics, an extension topology is a topology placed on the disjoint union of a topological space and another set.There are various types of extension topology, described in the sections below.- Extension topology :...
    
    
13. Finite excluded point topology
    Excluded point topology
    In mathematics, the excluded point topology is a topology where exclusion of a particular point defines openness. Formally, let X be any set and p ∈ X. The collectionof subsets of X is then the excluded point topology on X....
    
    
14. Countable excluded point topology
    Excluded point topology
    In mathematics, the excluded point topology is a topology where exclusion of a particular point defines openness. Formally, let X be any set and p ∈ X. The collectionof subsets of X is then the excluded point topology on X....
    
    
15. Uncountable excluded point topology
    Excluded point topology
    In mathematics, the excluded point topology is a topology where exclusion of a particular point defines openness. Formally, let X be any set and p ∈ X. The collectionof subsets of X is then the excluded point topology on X....
    
    
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
    Fort space
    In mathematics, Fort space, named after M. K. Fort, Jr., is an example in the theory of topological spaces.Let X be an infinite set of points, of which P is one...
    
    
24. Uncountable Fort space
    Fort space
    In mathematics, Fort space, named after M. K. Fort, Jr., is an example in the theory of topological spaces.Let X be an infinite set of points, of which P is one...
    
    
25. Fortissimo space
26. Arens–Fort space
27. Modified Fort space
    Fort space
    In mathematics, Fort space, named after M. K. Fort, Jr., is an example in the theory of topological spaces.Let X be an infinite set of points, of which P is one...
    
    
28. Euclidean topology
    Euclidean space
    In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
    
    
29. Cantor set
    Cantor set
    In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....
    
    
30. Rational number
    Rational number
    In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
    
    s
31. Irrational number
    Irrational number
    In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
    
    s
32. Special subsets of the real line
33. Special subsets of the plane
34. One point compactification
    Compactification
    Compactification may refer to:* Compactification , making a topological space compact* Compactification , the "curling up" of extra dimensions in string theory* Compaction...
    
     topology
35. One point compactification of the rationals
36. Hilbert space
    Hilbert space
    The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
    
    
37. Fréchet space
    Fréchet space
    In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces...
    
    
38. Hilbert cube
    Hilbert cube
    In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology...
    
    
39. Order topology
    Order topology
    In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets...
    
    
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
    Long line (topology)
    In topology, the long line is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties. Therefore it serves as one of the basic counterexamples of topology...
    
    
46. Extended long line
    Long line (topology)
    In topology, the long line is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties. Therefore it serves as one of the basic counterexamples of topology...
    
    
47. An altered long line
    Long line (topology)
    In topology, the long line is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties. Therefore it serves as one of the basic counterexamples of topology...
    
    
48. Lexicographic order topology on the unit square
    Lexicographic order topology on the unit square
    In mathematics, and especially general topology, the lexicographic ordering on the unit square is an example of a topology on the unit square S, i.e...
    
    
49. Right order topology
    Order topology
    In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets...
    
    
50. Right order topology on R
    Order topology
    In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets...
    
    
51. Right half-open interval topology
    Lower limit topology
    In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set R of real numbers; it is different from the standard topology on R and has a number of interesting properties...
    
    
52. Nested interval topology
    Nested interval topology
    In mathematics, more specifically general topology, the nested interval topology is an example of a topology given to the open interval , i.e. the set of all real numbers x such that...
    
    
53. Overlapping interval topology
    Overlapping interval topology
    In mathematics, the overlapping interval topology is a topology which is used to illustrate various topological principles.-Definition:Given the closed interval [-1,1] of the real number line, the open sets of the topology are generated from the half-open intervals [-1,b) and In mathematics, the...
    
    
54. Interlocking interval topology
    Interlocking interval topology
    In mathematics, and especially general topology, the interlocking interval topology is an example of a topology on the set , i.e. the set of all positive real numbers that are not positive whole numbers...
    
    
55. Hjalmar Ekdal topology
    Hjalmar Ekdal topology
    In mathematics, the Hjalmar Ekdal topology is a special example in the theory of topological spaces.The Hjalmar Ekdal topology consists of N* together with the collection of all subsets of N* in which every odd member is accompanied by its even successor...
    
    
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
    Rational sequence topology
    In mathematics, more specifically general topology, the rational sequence topology is an example of a topology given to the set of real numbers, denoted R....
    
    
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
    Double origin topology
    In mathematics, more specifically general topology, the double origin topology is an example of a topology given to the plane R2 with an extra point, say 0*, added...
    
    
75. Irrational slope topology
76. Deleted diameter topology
77. Deleted radius topology
78. Half-disk topology
    Half-disk topology
    In mathematics, and particularly general topology, the half-disk topology is an example of a topology given to the set X, given by all points in the plane such that...
    
    
79. Irregular lattice topology
80. Arens square
81. Simplified Arens square
82. Niemytzki's tangent disk topology
    Moore plane
    In mathematics, the Moore plane, also sometimes called Niemytzki plane is a topological space. It is a completely regular Hausdorff space which is not normal...
    
    
83. Metrizable tangent disk topology
84. Sorgenfrey's half-open square topology
85. Michael's product topology
86. Tychonoff plank
    Tychonoff plank
    In topology, the Tychonoff plank is a topological space that is a counterexample to several plausible-sounding conjectures. It is defined as the product of the two ordinal space[0,\omega_1]\times[0,\omega]...
    
    
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. Z^Z
103. Uncountable products of Z⁺
104. Baire product metric on R^ω
105. I^I
106. [0,Ω)×I^I
107. Helly space
     Helly space
     In mathematics, and particularly functional analysis, the Helly space, named after Eduard Helly, consists of all monotonically increasing functions , where [0,1] denotes the closed interval given by the set of all x such that In other words, for all we have and also if then Let the closed...
     
     
108. C[0,1]
109. Box product topology on R^ω
110. Stone–Čech compactification
     Stone–Cech compactification
     In the mathematical discipline of general topology, Stone–Čech compactification is a technique for constructing a universal map from a topological space X to a compact Hausdorff space βX...
     
     
111. Stone–Čech compactification
     Stone–Cech compactification
     In the mathematical discipline of general topology, Stone–Čech compactification is a technique for constructing a universal map from a topological space X to a compact Hausdorff space βX...
     
      of the integers
112. Novak space
113. Strong ultrafilter topology
114. Single ultrafilter topology
115. Nested rectangles
116. Topologist's sine curve
     Topologist's sine curve
     In the branch of mathematics known as topology, the topologist's sine curve is a topological space with several interesting properties that make it an important textbook example....
     
     
117. Closed topologist's sine curve
     Topologist's sine curve
     In the branch of mathematics known as topology, the topologist's sine curve is a topological space with several interesting properties that make it an important textbook example....
     
     
118. Extended topologist's sine curve
     Topologist's sine curve
     In the branch of mathematics known as topology, the topologist's sine curve is a topological space with several interesting properties that make it an important textbook example....
     
     
119. Infinite broom
     Infinite broom
     In topology, the infinite broom is a subset of the Euclidean plane that is used as an example distinguishing various notions of connectedness. The closed infinite broom is the closure of the infinite broom, and is also referred to as the broom space.-Definition:The infinite broom is the subset of...
     
     
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
     Knaster-Kuratowski fan
     In topology, a branch of mathematics, the Knaster–Kuratowski fan is a connected topological space with the property that the removal of a single point makes it totally disconnected.Let C be the Cantor set, let p be the point \in\mathbb R^2, and...
     
     
129. Cantor's teepee
130. Pseudo-arc
     Pseudo-arc
     In general topology, the pseudo-arc is the simplest nondegenerate hereditarily indecomposable continuum. Pseudo-arc is an arc-like homogeneous continuum. R.H...
     
     
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
     Hausdorff distance
     In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a metric space in its own right...
     
      topology
139. Post Office metric
140. Radial metric
141. Radial interval topology
142. Bing's discrete extension space
143. Michael's closed subspace