For example, the word "nth" is an example of an English word without a vowel. It is a counterexample to the claim that every English word contains a vowel. In the opinion of B. Gelbaum and J. Olmsted - the authors of two popular books on counterexamples - much of mathematical development consists in finding and proving theorems and counterexamples. This book is intended for intermediate readers.
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 , 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.
A MSE Collection: Topology Book References
This particular counterexample shows that second-countability does not follow from first-countability. In her review of the first edition, Mary Ellen Rudin wrote:. In his submission  to Mathematical Reviews C. Wayne Patty wrote:. Munkres, 2nd edition, in the sense that they cover the same material as does Munkres; prove the same theorems as are proved in Munkres, but filling in the details omitted by Munkres; use the same definitions as used by Munkres; include as solved examples some, most, or all of Munkres' exercise problems?
Of course, one cannot expect a text to fulfill all the above requirements, but which one s do es this the best?
Aspects of Topology by C. Christenson and W.
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Voxman is an easy to read, instructive text about general topology containing a lot of nice graphics. This AMS review might be useful.
Counterexamples in Topology by L. Steen and L.
This is a great resource to look for topological spaces having specific properties and to look for topological properties and their relationship. It contains extensive charts of terms like compactness and the relationship of their different flavors.
See also this Wikipage. I'm fond of Introduction to Topological Manifolds by Lee.
- Counterexamples in Topology - Wikipedia.
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It's about halfway between Munkres and Hatcher in terms of content, and filled with examples. It also acts as a good prologue for his Introduction to Smooth Manifolds , if you want to see what differential geometry has to offer.
Counterexamples in Topology
The Schaum's outline of General Topology is the best book ever to learn Topology from Munkres' is difficult to learn from, because like most American Math Texts, it does not have enough worked out examples This is a rather difficult subject to learn without a good professor guiding the learner However one small advantage that Munkres' has is that one can find solutions to some of the exercises online because it is a widely used book , so you can check your work sometimes if you are doing the exercises by yourself I would not recommend the other books you mention I really liked the book Elementary Topology.
It is based on topology classes at the Faculty of Mathematics and Mechanics of the Leningrad State University in the s. As the title suggests, the greatest portion of the book is made up of problems, exercises and examples, which are great for absorbing the material.
Furthermore, the online version is available for free in the author's webpage. Note that the online version does not include proofs or solutions to exercises , but it's great as a problem book.