Counterexamples in Topology
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Counterexamples in Topology (1978) 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
- 21. Double pointed 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
- 116. Topologist's sine curve
- 117. Closed topologist's sine curve
- 118. Extended topologist's sine curve
See also
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).