Counterexamples in Topology (1970) is a mathematics book by topologists Lynn A. Steen and J. Arthur Seebach, Jr. .

Together with their graduate students, Steen and Seebach canvassed the field of topology for a wide grouping of topological counterexamples. If you're wondering whether one property of topological spaces follows from another, this book can usually provide a counterexample if it's false. For example, is there an example of a first-countable space which is not second-countable? The answer is yes, as counterexample #3 (the discrete topology on an uncountable set) is the first to show.

Steen and Seebach were working on the metrization problem, which asks which topological spaces can be made into metric spaces. This problem had inspired topologists to define a wealth of topological properties, some of which metric spaces had and some of which they did not. By comparing and contrasting these properties in a single reference, Steen and Seebach simplified the relevant literature. Several other "Counterexamples in ..." books and papers have followed.

Note that several of the naming conventions in this book differ from more modern conventions (including those in Wikipedia), particularly with respect to the separation axioms. Steen and Seebach exchange the meanings of T₃, T₄, and T₅ with those of regular, normal, and completely normal. They also exchange the meanings of completely Hausdorff with Urysohn. This was a result of the different historical development of metrization theory and general topology; see History of the separation axioms for more.

Links to counterexamples

1. Finite discrete topology

2. Countable discrete topology

3. Uncountable discrete topology

4. Indiscrete topology

11. Sierpinski space

18. Finite complement topology on a countable space

19. Finite complement topology on an uncountable space

20. Countable complement topology

28. Euclidean topology

29. Cantor set

30. Rational numbers

31. Irrational numbers

36. Hilbert space

37. Fréchet space

38. Hilbert cube

39. Order topology

45. Long line

46. Extended long line

51. Right half-open interval topology

84. Sorgenfrey's half-open square topology

116. Topologist's sine curve

117. Closed topologist's sine curve

118. Extended topologist's sine curve

See also

• List of examples in general topology

References

• 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).