Counterexamples in Topology

Counterexamples in Topology Author(s) Lynn Arthur Steen
J. Arthur Seebach, Jr.Country United States Language English Subject(s) Topological spaces Publisher SpringerVerlag Publication date 1970 Media type Hardback, Paperback Pages 244pp. ISBN ISBN 048668735X
(Dover edition)OCLC Number 32311847 Dewey Decimal 514/.3 20 LC Classification QA611.3 .S74 1995 Counterexamples in Topology (1970, 2nd ed. 1978) is a book on mathematics by topologists 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 spaces 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 firstcountable space which is not secondcountable is counterexample #3, the discrete topology on an uncountable set. This particular counterexample shows that secondcountability does not follow from firstcountability.
Several other "Counterexamples in ..." books and papers have followed, with similar motivations.
Contents
Notation
Several of the naming conventions in this book differ from more accepted modern conventions, particularly with respect to the separation axioms. The authors use the terms T_{3}, T_{4}, and T_{5} to refer to regular, normal, and completely normal. They also refer to completely Hausdorff as Urysohn. This was a result of the different historical development of metrization theory and general topology; see History of the separation axioms for more.
List of mentioned counterexamples
 Finite discrete topology
 Countable discrete topology
 Uncountable discrete topology
 Indiscrete topology
 Partition topology
 Odd–even topology
 Deleted integer topology
 Finite particular point topology
 Countable particular point topology
 Uncountable particular point topology
 Sierpinski space, see also particular point topology
 Closed extension topology
 Finite excluded point topology
 Countable excluded point topology
 Uncountable excluded point topology
 Open extension topology
 Eitheror 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 topology
 Cantor set
 Rational numbers
 Irrational numbers
 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
 Extended long line
 An altered long line
 Lexicographic order topology on the unit square
 Right order topology
 Right order topology on R
 Right halfopen interval topology
 Nested interval topology
 Overlapping interval topology
 Interlocking interval topology
 Hjalmar Ekdal topology
 Prime ideal topology
 Divisor topology
 Evenly spaced integer topology
 The padic topology on Z
 Relatively prime integer topology
 Prime integer topology
 Double pointed reals
 Countable complement extension 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
 Halfdisk topology
 Irregular lattice topology
 Arens square
 Simplified Arens square
 Niemytzki's tangent disk topology
 Metrizable tangent disk topology
 Sorgenfrey's halfopen 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
 Weak parallel line topology
 Strong parallel line topology
 Concentric circles
 Appert space
 Maximal compact topology
 Minimal Hausdorff topology
 Alexandroff square
 Z^{Z}
 Uncountable products of Z^{+}
 Baire product metric on R^{ω}
 I^{I}
 [0,Ω)×I^{I}
 Helly space
 C[0,1]
 Box product topology on R^{ω}
 Stone–Čech compactification
 Stone–Čech compactification of the integers
 Novak space
 Strong ultrafilter topology
 Single ultrafilter topology
 Nested rectangles
 Topologist's sine curve
 Closed 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
 Cantor's leaky tent
 Cantor's teepee
 Pseudoarc
 Miller's biconnected set
 Wheel without its hub
 Tangora's connected space
 Bounded metrics
 Sierpinski's metric space
 Duncan's space
 Cauchy completion
 Hausdorff's metric topology
 Post Office metric
 Radial metric
 Radial interval topology
 Bing's discrete extension space
 Michael's closed subspace
See also
References
 Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. SpringerVerlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 048668735X (Dover edition).
Categories: General topology
 1978 books
 Mathematics books
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