Review of the book
Counterexamples in Topology
by Lynn Arthur Steen and J. Arthur Seebach, Jr.
The first part of this book contains basic definitions from General Topology and discusses the logical relationships between these concepts: after giving a number of related definitions it discusses which implications hold and which don't. This can serve as a source of topological terminology - a kind of topological encyclopedic dictionary with references to concrete examples.
The second part contains examples of topological spaces and comments on their properties. These examples are cited in the first part to illustrate that certain implications do not hold in general. There are over 140 examples of concrete topological spaces. Most of them are very complicated and unintuitive. It may be enlightening (or frightening) to even browse through these examples without actually reading them with full attention.
One of the most useful parts of this book is the appendix titled General Reference Chart, which contains a large table extending over ten pages. The columns are labeled with all the topological properties discussed in the book. The rows are labeled with the numbers of all the example topological spaces listed in the book. The cells of this table contain zeros and ones. "1" means that the example spcae has the indicated property and "0" means that the space satisfies the negation of that property.
If you have a particular example space in mind you can use this table to determine exactly which properties the space satisfies and which it fails to satisfy. Alternately, if you have in mind a particular combination of topological properties you can locate example spaces with these properties (if they are listed in this book) or you can start suspecting that such spaces don't exist.
This reference chart can be used to find counterexamples for false propositions or to formulate a candidate for a general theorem. For example, if you wonder whether every locally compact Hausdorff space must be normal you can look for the combination of T_2, not(T_4), and locally compact. You will then find a couple of examples of such spaces. And if you look carefully you will notice that all these examples are s-compact (countable unions of compact sets) and thus you may formulate the conjecture that every s-compact locally compact Hausdorff space must be normal.
See the table of contents of this book.