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

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

topologist In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...

Lynn Steen Lynn Arthur Steen (January 1, 1941 – June 21, 2015) was an American mathematician who was a Professor of Mathematics at St. Olaf College, Northfield, Minnesota in the U.S. He wrote numerous books and articles on the teaching of mathematics. H ...

and

J. Arthur Seebach, Jr. J. Arthur Seebach Jr (May 17, 1938 – December 3, 1996) was an American mathematician. Seebach studied Greek language as an undergraduate, making it a second major with mathematics. Seebach studied with A. I. Weinzweig at Northwestern Univ ...

In the process of working on problems like the metrization problem, topologists (including Steen and Seebach) have defined a wide variety of

topological properties In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spac ...

. It is often useful in the study and understanding of abstracts such as

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...

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

counterexample A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...

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,

Minnesota Minnesota () is a state in the upper midwestern region of the United States. It is the 12th largest U.S. state in area and the 22nd most populous, with over 5.75 million residents. Minnesota is home to western prairies, now given over to ...

in the summer of 1967, canvassed the field of

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

for such counterexamples and compiled them in an attempt to simplify the literature. For instance, an example of a

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 (local base) ...

which is not

second-countable In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...

is counterexample #3, the

discrete topology In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...

on an

uncountable set In mathematics, an uncountable set (or uncountably infinite 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 numb ...

. 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

In her review of the first edition,

Mary Ellen Rudin Mary Ellen Rudin (December 7, 1924 – March 18, 2013) was an American mathematician known for her work in set-theoretic topology. In 2013, Elsevier established the Mary Ellen Rudin Young Researcher Award, which is awarded annually to a young res ...

wrote: :In other mathematical fields one restricts one's problem by requiring that the

space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...

be Hausdorff or

paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, ...

metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...

, 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... In his submission to

Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also pu ...

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 in

Advances in Mathematics ''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed ...

treated topology as territory to be explored: :

Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...

once said that every mathematician should be something of a naturalist. 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.

Notation

Several of the

naming convention A naming convention is a convention (generally agreed scheme) for naming things. Conventions differ in their intents, which may include to: * Allow useful information to be deduced from the names based on regularities. For instance, in Manhatta ...

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

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. These are sometimes ...

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

normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...

, and

completely normal 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. Th ...

. They also refer to

completely Hausdorff In topology, a discipline within mathematics, 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 t ...

Urysohn Pavel Samuilovich Urysohn () (February 3, 1898 – August 17, 1924) was a Soviet mathematician who is best known for his contributions in dimension theory, and for developing Urysohn's metrization theorem and Urysohn's lemma, both of which are f ...

. This was a result of the different historical development of metrization theory and

general topology In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...

; see

History of the separation axioms The history of the separation axioms in general topology has been convoluted, with many meanings competing for the same terms and many terms competing for the same concept. Origins Before the current general definition of topological space, th ...

for more. The

long line Long line or longline may refer to: *'' Long Line'', an album by Peter Wolf * Long line (topology), or Alexandroff line, a topological space *Long line (telecommunications), a transmission line in a long-distance communications network *Longline fi ...

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

List of mentioned counterexamples

Finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...

Countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...

#Uncountable

Indiscrete topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...

Partition topology In mathematics, the partition topology is a topological space, topology that can be induced on any set X by Partition of a set, partitioning X into disjoint subsets P; these subsets form the basis (topology), basis for the topology. There are two i ...

# Odd–even topology # Deleted integer topology # Finite particular point topology # Countable particular point topology # Uncountable particular point topology #

Sierpiński space In mathematics, the Sierpiński space (or the connected two-point set) 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. It is named ...

, see also

particular point topology In mathematics, the particular point topology (or included point topology) is a topology where a set is open if it contains a particular point of the topological space. Formally, let ''X'' be any non-empty set and ''p'' ∈ ''X''. The collec ...

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 Le ...

#Finite

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 non-empty set and ''p'' ∈ ''X''. The collection :T = \ \cup \ of subsets of ''X'' is then the excluded ...

#Countable

#Uncountable

# Open extension topology #

Either-or topology ''Either/Or'' ( Danish: ''Enten – Eller'') is the first published work of the Danish philosopher Søren Kierkegaard. Appearing in two volumes in 1843 under the pseudonymous editorship of ''Victor Eremita'' (Latin for "victorious hermit"), it ...

Finite complement topology In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is cocou ...

on a

countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...

space #

on an uncountable space #

Countable complement topology The cocountable topology or countable complement topology on any set ''X'' consists of the empty set and all cocountable subsets of ''X'', that is all sets whose Complement (set theory), complement in ''X'' is countable set, countable. It follows th ...

#Double pointed

countable complement topology The cocountable topology or countable complement topology on any set ''X'' consists of the empty set and all cocountable subsets of ''X'', that is all sets whose Complement (set theory), complement in ''X'' is countable set, countable. It follows th ...

# Compact complement topology #Countable

Fort space In mathematics, there are a few topological spaces named after M. K. Fort, Jr. Fort space Fort space is defined by taking an infinite set ''X'', with a particular point ''p'' in ''X'', and declaring open the subsets ''A'' of ''X'' such that: * ...

#Uncountable

# Fortissimo space #

Arens–Fort space In mathematics, the Arens–Fort space is a special example in the theory of topological spaces, named for Richard Friederich Arens and M. K. Fort, Jr. Definition The Arens–Fort space is the topological space (X,\tau) where X is the set of o ...

#Modified

Euclidean topology In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean distance, Euclidean metric. Definition The Euclidean norm on \R^n is the non-negative f ...

Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. Thr ...

Rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...

s #

Irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...

s #Special subsets of the real line #Special subsets of the plane # One point compactification topology #One point compactification of the rationals #

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...

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 ( normed vector spaces that are complete with respect to th ...

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. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, c ...

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. If ''X'' is a totally ordered set, t ...

#Open ordinal space ,Γ)_where_Γ<Ω #Closed_ordinal_space_[0,Γwhere_Γ<Ω #Open_ordinal_space_

,Γ)_where_Γ<Ω #Closed_ordinal_space_[0,Γwhere_Γ<Ω #Open_ordinal_space_[0,Ω) #Closed_ordinal_space_[0,Ω">,Γ.html"_;"title=",Γ)_where_Γ<Ω #Closed_ordinal_space_[0,Γ">,Γ)_where_Γ<Ω #Closed_ordinal_space_[0,Γwhere_Γ<Ω #Open_ordinal_space_[0,Ω) #Closed_ordinal_space_[0,Ω#Uncountable_discrete_ordinal_space # ,Γ)_where_Γ<Ω #Closed_ordinal_space_[0,Γwhere_Γ<Ω #Open_ordinal_space_[0,Ω) #Closed_ordinal_space_[0,Ω">,Γ.html"_;"title=",Γ)_where_Γ<Ω #Closed_ordinal_space_[0,Γ">,Γ)_where_Γ<Ω #Closed_ordinal_space_[0,Γwhere_Γ<Ω #Open_ordinal_space_[0,Ω) #Closed_ordinal_space_[0,Ω#Uncountable_discrete_ordinal_space #Long_line_(topology)">Long_line_Long_line_or_longline_may_refer_to: *''_Long_Line'',_an_album_by_Peter_Wolf *_Long_line_(topology),_or_Alexandroff_line,_a_topological_space *Long_line_(telecommunications),_a_transmission_line_in_a_long-distance_communications_network *Longline_fi_...

#Long_line_(topology)">Extended_long_line #An_altered_Long_line_(topology).html" "title="ong_line_(topology).html" ;"title="Long_line_(topology).html" "title=",Ω) #Closed_ordinal_space_[0,Ω.html" ;"title=",Γ.html" ;"title=",Γ) where Γ<Ω #Closed ordinal space [0,Γ">,Γ) where Γ<Ω #Closed ordinal space [0,Γwhere Γ<Ω #Open ordinal space [0,Ω) #Closed ordinal space [0,Ω">,Γ.html" ;"title=",Γ) where Γ<Ω #Closed ordinal space [0,Γ">,Γ) where Γ<Ω #Closed ordinal space [0,Γwhere Γ<Ω #Open ordinal space [0,Ω) #Closed ordinal space [0,Ω#Uncountable discrete ordinal space #Long line (topology)">Long line Long line or longline may refer to: *'' Long Line'', an album by Peter Wolf * Long line (topology), or Alexandroff line, a topological space *Long line (telecommunications), a transmission line in a long-distance communications network *Longline fi ...

Reviews

Notation

List of mentioned counterexamples

See also

References

External links