Counterexamples in Topology

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 first-countable space which is not 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 space be Hausdorff or paracompact or 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...

In his submission 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:
 * Lebesgue 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 conventions in this book differ from more accepted modern conventions, particularly with respect to the separation axioms. The authors use the terms T3, T4, and T5 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.

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

List of mentioned 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) Sierpiński space, see also particular point topology
 * 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) Cantor set
 * 30) Rational numbers
 * 31) 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 [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
 * 46) Extended long line
 * 47) An altered long line
 * 48) Lexicographic order topology 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, whose name was introduced in this book.
 * 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
 * 77) Deleted radius topology
 * 78) Half-disk topology
 * 79) Irregular lattice topology
 * 80) Arens square
 * 81) Simplified Arens square
 * 82) Niemytzki's tangent disk topology
 * 83) Metrizable tangent disk 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'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) ZZ
 * 103) Uncountable products of Z+
 * 104) Baire product metric on Rω
 * 105) II
 * [0,Ω)×II
 * 1) Helly space
 * 2) C[0,1]
 * 3) Box product topology on Rω
 * 4) Stone–Čech compactification
 * 5) Stone–Čech compactification of the integers
 * 6) Novak space
 * 7) Strong ultrafilter topology
 * 8) Single ultrafilter topology
 * 9) Nested rectangles
 * 10) Topologist's sine curve
 * 11) Closed topologist's sine curve
 * 12) Extended topologist's sine curve
 * 13) Infinite broom
 * 14) Closed infinite broom
 * 15) Integer broom
 * 16) Nested angles
 * 17) Infinite cage
 * 18) Bernstein's connected sets
 * 19) Gustin's sequence space
 * 20) Roy's lattice space
 * 21) Roy's lattice subspace
 * 22) Cantor's leaky tent
 * 23) Cantor's teepee
 * 24) Pseudo-arc
 * 25) Miller's biconnected set
 * 26) Wheel without its hub
 * 27) Tangora's connected space
 * 28) Bounded metrics
 * 29) Sierpinski's metric space
 * 30) Duncan's space
 * 31) Cauchy completion
 * 32) Hausdorff's metric topology
 * 33) Post Office metric
 * 34) Radial metric
 * 35) Radial interval topology
 * 36) Bing's discrete extension space
 * 37) Michael's closed subspace