Editing Counterexamples in Topology

{{short description|Book by Lynn Steen}}
{{Infobox book 
| name          = Counterexamples in Topology
| image         = Counterexamples in Topology.jpg
| caption       = 
| author        = [[Lynn Steen|Lynn Arthur Steen]]<br>[[J. Arthur Seebach, Jr.]]
| country       = United States
| language      = English
| series        = 
| genre         = Non-fiction
| subject       = [[Topological space]]s
| publisher     = [[Springer-Verlag]]
| release_date  = 1970
| media_type    = [[Hardback]], [[Paperback]]
| pages         = 244 pp.
| isbn =  0-486-68735-X
| dewey= 514/.3 20
| congress= QA611.3 .S74 1995
| oclc= 32311847
}}

'''''Counterexamples in Topology''''' (1970, 2nd ed. 1978) is a book on [[mathematics]] by [[topology|topologist]]s [[Lynn Steen]] and [[J. Arthur Seebach, Jr.]]

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]]s to determine that one property does not follow from another.  One of the easiest ways of doing this is to find a [[counterexample]] 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]] in the summer of 1967, canvassed the field of [[topology]] for such counterexamples and compiled them in an attempt to simplify the literature.

For instance, an example of a [[first-countable space]] which is not [[second-countable space|second-countable]] is counterexample #3, the [[discrete topology]] on an [[uncountable set]].  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]] wrote:
:In other mathematical fields one restricts one's problem by requiring that the [[topological space|space]] be [[Hausdorff space|Hausdorff]] or [[paracompact]] or [[metric space|metric]], 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...<ref>{{cite journal | first=Mary Ellen | last=Rudin | year=1971 | title=Review: ''Counterexamples in Topology'' | journal=[[American Mathematical Monthly]] | doi=10.2307/2318037 | volume=78 | issue=7 | pages=803–804 | jstor=2318037 | mr=1536430}}</ref>
In his submission<ref>C. Wayne Patty (1971) "Review: ''Counterexamples in Topology''", {{MR|id=0266131}}</ref> to [[Mathematical Reviews]] 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]] treated topology as territory to be explored:
:[[Henri Lebesgue|Lebesgue]] once said that every mathematician should be something of a [[Naturalism_(philosophy)|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.<ref>{{cite journal | first1=Joseph | last1=Kung | first2=Gian-Carlo | last2=Rota | authorlink2=Gian-Carlo Rota | year=1979 | title=Review: ''Counterexamples in Topology'' | journal=[[Advances in Mathematics]] | volume=32 | issue=1 | pages=81 | doi=10.1016/0001-8708(79)90031-8 | doi-access=free}}</ref>

==Notation==
Several of the [[naming convention]]s in this book differ from more accepted modern conventions, particularly with respect to the [[separation axiom]]s. The authors use the terms T<sub>3</sub>, T<sub>4</sub>, and T<sub>5</sub> to refer to [[regular space|regular]], [[normal space|normal]], and [[completely normal space|completely normal]]. They also refer to  [[completely Hausdorff space|completely Hausdorff]] as [[Urysohn and completely Hausdorff spaces|Urysohn]]. This was a result of the different historical development of metrization theory and [[general topology]]; see [[History of the separation axioms]] for more.

The [[long line (topology)|long line]] in example 45 is what most topologists nowadays would call the 'closed long ray'.

==List of mentioned counterexamples==
{{div col|colwidth=30em}}
#[[finite set|Finite]] [[discrete topology]]
#[[Countable]] [[discrete topology]]
#Uncountable [[discrete topology]]
#[[Indiscrete topology]]
#[[Partition topology]]
#[[Odd–even topology]]
#[[Deleted integer topology]]
#[[Particular point topology|Finite particular point topology]]
#[[Particular point topology|Countable particular point topology]]
#[[Particular point topology|Uncountable particular point topology]]
#[[Sierpiński space]], see also [[particular point topology]]
#[[Closed extension topology]]
#Finite [[excluded point topology]]
#Countable [[excluded point topology]]
#Uncountable [[excluded point topology]]
#[[Open extension topology]]
#[[Either-or topology]]
#[[Finite complement topology]] on a [[countable]] space
#[[Finite complement topology]] on an uncountable space
#[[Countable complement topology]]
#Double pointed [[countable complement topology]]
#[[Compact complement topology]]
#Countable [[Fort space]]
#Uncountable [[Fort space]]
#[[Fortissimo space]]
#[[Arens–Fort space]]
#Modified [[Fort space]]
#[[Euclidean space|Euclidean topology]]
#[[Cantor set]]
#[[Rational number]]s
#[[Irrational number]]s
#Special subsets of the real line
#Special subsets of the plane
#[[One point compactification]] topology
#One point compactification of the rationals
#[[Hilbert space]]
#[[Fréchet space]]
#[[Hilbert cube]]
#[[Order topology]]
#Open ordinal space [0,Γ) where Γ<Ω
#Closed ordinal space [0,Γ] where Γ<Ω
#Open ordinal space [0,Ω)
#Closed ordinal space [0,Ω]
#Uncountable discrete ordinal space
#[[Long line (topology)|Long line]]
#[[Long line (topology)|Extended long line]]
#An altered [[Long line (topology)|long line]]
#[[Lexicographic order topology on the unit square]]
#[[Order topology|Right order topology]]
#[[Order topology|Right order topology on '''R''']]
#[[Lower limit topology|Right half-open interval topology]]
#[[Nested interval topology]]
#[[Overlapping interval topology]]
#[[Interlocking interval topology]]
#Hjalmar Ekdal topology, whose name was introduced in this book.
#[[Prime ideal topology]]
#[[Divisor topology]]
#[[Evenly spaced integer topology]]
#The [[P-adic#Analytic approach|''p''-adic topology]] on '''Z'''
#Relatively [[prime integer topology]]
#[[Prime integer topology]]
#Double pointed reals
#Countable complement extension topology
#[[K-topology|Smirnov's deleted sequence topology]]
#[[Rational sequence topology]]
#Indiscrete rational extension of '''R'''
#Indiscrete irrational extension of '''R'''
#Pointed rational extension of '''R'''
#Pointed irrational extension of '''R'''
#Discrete rational extension of '''R'''
#Discrete irrational extension of '''R'''
#Rational extension in the plane
#[[Telophase topology]]
#[[Double origin topology]]
# [[Irrational slope topology]]
#Deleted diameter topology
#Deleted radius topology
#[[Half-disk topology]]
#Irregular lattice topology
#[[Arens square]]
#Simplified [[Arens square]]
#[[Moore plane|Niemytzki's tangent disk topology]]
#Metrizable tangent disk topology
#[[Sorgenfrey plane|Sorgenfrey's half-open square topology]]
#Michael's product topology
#[[Tychonoff plank]]
#[[Deleted Tychonoff plank]]
#[[Alexandroff plank]]
#[[Dieudonné plank]]
#Tychonoff corkscrew
#Deleted Tychonoff corkscrew
#[[Hewitt's condensed corkscrew]]
#[[Thomas's plank]]
#[[Thomas's corkscrew]]
#[[Split interval|Weak parallel line topology]]
#Strong parallel line topology
#Concentric circles
#[[Appert space]]
#Maximal compact topology
#Minimal [[Hausdorff topology]]
#[[Alexandroff square]]
#'''Z<sup>Z</sup>'''
#Uncountable products of '''Z'''<sup>+</sup>
#Baire product metric on '''R'''<sup>ω</sup>
#'''I<sup>I</sup>'''
#[0,Ω)×'''I<sup>I</sup>'''
#[[Helly space]]
#'''C'''[0,1]
#[[Box topology|Box product topology]] on '''R'''<sup>ω</sup>
#[[Stone–Čech compactification]]
#[[Stone–Čech compactification]] of the integers
#[[Novak space]]
#Strong ultrafilter topology
#Single ultrafilter topology
#Nested rectangles
#[[Topologist's sine curve]]
#[[Topologist's sine curve|Closed topologist's sine curve]]
#[[Topologist's sine curve|Extended topologist's sine curve]]
#[[Infinite broom]]
#[[Closed infinite broom]]
#[[Integer broom]]
#Nested angles
#Infinite cage
#[[Bernstein's connected sets]]
#[[Gustin's sequence space]]
#[[Roy's lattice space]]
#[[Roy's lattice subspace]]
#[[Knaster-Kuratowski fan|Cantor's leaky tent]]
#[[Cantor's teepee]]
#[[Pseudo-arc]]
#[[Miller's biconnected set]]
#Wheel without its hub
#[[Tangora's connected space]]
#Bounded metrics
#[[Sierpinski's metric space]]
#[[Duncan's space]]
#[[Cauchy completion]]
#[[Hausdorff distance|Hausdorff's metric]] topology
#[[Post Office metric]]
#Radial metric
#Radial interval topology
#[[Bing's discrete extension space]]
#[[Michael's closed subspace]]
{{div col end}}

==See also==

* {{annotated link|List of examples in general topology}}

==References==
{{reflist}}

==Bibliography==
* {{cite book|last1=Steen|first1=Lynn Arthur|author-link1=Lynn Arthur Steen|last2=Seebach|first2=J. Arthur|author-link2=J. Arthur Seebach Jr.|title=[[Counterexamples in topology]]|publisher=Springer New York|publication-place=New York, NY|year=1978|isbn=978-0-387-90312-5|doi=10.1007/978-1-4612-6290-9}} <!--{{sfn|Steen|Seebach|1978|p=}}-->
* {{cite book|last1=Steen|first1=Lynn Arthur|author-link1=Lynn Arthur Steen|last2=Seebach|first2=J. Arthur|author-link2=J. Arthur Seebach Jr.|title=[[Counterexamples in topology]]|publisher=Dover Publications|publication-place=New York|date=1995|orig-date=First published 1978 by Springer-Verlag, New York|isbn=0-486-68735-X|oclc=32311847}} <!--{{sfn|Steen|Seebach|1995|p=}}-->
*Lynn Arthur Steen and J. Arthur Seebach, Jr., ''Counterexamples in Topology''. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. {{ISBN|0-486-68735-X}} (Dover edition).

==External links==

* [http://topology.pi-base.org/ π-Base: An Interactive Encyclopedia of Topological Spaces]

[[Category:1978 non-fiction books]]
[[Category:Books about mathematical counterexamples|Topology]]
[[Category:General topology]]
[[Category:Mathematics books]]