Counterexamples in Topology

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.

This book contains a dense covering of point-set topology and over a hundred different topologies over different spaces (as per the book count; some of those include more than one topology over the same space, or one topology over several spaces.) You can learn a great deal about topology just from this book without help. While there are no exercises and no proofs, there are plenty of examples to show why one property of a topology is or is not dependent on another. Thinking your way through the introduction and examples is a great way to learn.

On the down side, there's a topology paper included as an appendix that has little to do with the book and seems to be included just as an important paper on topology. It's way above my head, and show more I suspect it will be for many years. Also, the exercises seem randomly ordered, which is less of a problem because they are heavily linked by number from all over the book. show less

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?).