Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Counterexamples in Topology
Summary
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.
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.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.