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Counterexamples in Topology

Counterexamples in Topology (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr.

Author

English

Non-fiction

1970

United States

244 pp.

514/.3 20

QA611.3 .S74 1995

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.

Notation[edit]

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 examples in general topology

; Seebach, J. Arthur (1978). Counterexamples in topology. New York, NY: Springer New York. doi:10.1007/978-1-4612-6290-9. ISBN 978-0-387-90312-5.

Steen, Lynn Arthur

; Seebach, J. Arthur (1995) [First published 1978 by Springer-Verlag, New York]. Counterexamples in topology. New York: Dover Publications. ISBN 0-486-68735-X. OCLC 32311847.

Steen, Lynn Arthur

Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995.  0-486-68735-X (Dover edition).

ISBN

π-Base: An Interactive Encyclopedia of Topological Spaces