Counterexamples in Topology

by Lynn Arthur Steen, J. Arthur Seebach

According to the authors of this highly useful compendium, focusing on examples is an extremely effective method of involving undergraduate mathematics students in actual research. It is only as a result of pursuing the details of each example that students experience a significant increment in topological understanding. With that in mind, Professors Steen and Seebach have assembled 143 examples in this book, providing innumerable concrete illustrations of definitions, theorems, and general methods of proof. Far from presenting all relevant examples, however, the book instead provides a fruitful context in which to ask new questions and seek new answers.Ranging from the familiar to the obscure, the examples are preceded by a succinct exposition of general topology and basic terminology and theory. Each example is treated as a whole, with a highly geometric exposition that helps readers comprehend the material. Over 25 Venn diagrams and reference charts summarize the properties of the examples and allow students to scan quickly for examples with prescribed properties. In addition, discussions of general methods of constructing and changing examples acquaint readers with the art of constructing counterexamples. The authors have included an extensive collection of problems and exercises, all correlated with various examples, and a bibliography of 140 sources, tracing each uncommon example to its origin.This revised and expanded second edition will be especially useful as a course supplement and reference work for students of general topology. Moreover, it gives the instructor the flexibility to design his own course while providing students with a wealth of historically and mathematically significant examples. 1978 edition.

Genres: Mathematics, Reference, Textbooks, Nonfiction, Physics, Geometry

365 pages, Kindle Edition

First published January 1, 1970

Reviews

[Jake · Rating 4 out of 5]

A great companion to Munkres to have.

[Review 1 - 05/05/17.]

[Ronald Lett · Rating 4 out of 5]

Topology is abstract enough that if you are learning the subject for the first time, and you are not constantly challenging yourself to come up with concrete applications and counterexamples, you will probably learn very little. If you find the requirements of a particular theorem to be a bit over-the-top and find yourself a few brain cells short of coming up with a proper counterexample to illuminate why the theorem is stated in that way, this book will be extremely useful. Even if you can always come up with one, some of these examples may be simpler or more illuminating. And at this price, there is no reason every mathematician should not have a copy.

[D. · Rating 5 out of 5]

Every student of topology should have this. Steen and Seebach provide instances to illustrate every distinction commonly made in topology (e.g. regular but not normal, T1 but not Hausdorff). In the latter part of the book the authors offer a thorough discussion of metrizability (under what conditions can a topological space be given a metric that "agrees" with its topology?).

[Liora Sophie · Rating 5 out of 5]

This is a great side-piece to peruse during your topology studies. Read it alongside Munkres. It's just what it sounds like - a big list of examples. It can enrich your topology skills and deepen your understanding. Also, it's jam-packed with fascinating stuff.

[Owen Tanner · Rating 5 out of 5]

Pretty perfect maths book. Does what it says on the tin. One thing I would recommend is that someone has built a database of topological spaces that includes most of these, so if you can't afford/your library doesn't have this book try this website:

https://topology.jdabbs.com/spaces

[psb · Rating 5 out of 5]

I might have debated whether to give this 4 or 5 stars
if it had been a +$50 math book, but at $10, "the choice
(function) is clear ..."

[Stephen · Rating 4 out of 5]

As the title states, this book provides counterexamples in topology (that you were probably too lazy to come up with). Some of the examples were very critical in understanding topology at the undergraduate level, but as one may suspect, most of them were just so esoteric to the point of absurdity.

[Nick Black · Rating 5 out of 5]

The Dover "Counterexamples in..." line is generally awesome; no exception here.

[Lulu]

Topology