Amsterdam: Elsevier (ISBN 0-444-50355-2/hbk). x, 526 p. $ 145.00; EUR 145.00 (2004).
General topology is a branch of mathematics which uses a particularly extensive terminology. This makes the present book valuable for anyone working in the subject.
In 120 articles the editors and their contributors give nothing less than a survey of the state of the art of general topology.
The contents of the book follow roughly the Mathematics Subject Classification also used by Zentralblatt MATH. That is, the book is divided into 10 sections, from A: Generalities, B: Basic constructions to H: Connections with other structures, with the additional sections J: Influencies of other fields and K: Connections with other fields. The sections again contain articles, e.g. a-01: Topological spaces, a-02: Modified open and closed sets, a-03 and a-04 Cardinal Functions, a-05: Convergence, and so on.
The articles, generally between 2 and 4 pages long, can be read as excellent short introductions to their topic. They usually begin with some – often historical – remarks putting the topic in context. Then follows a survey of the most important notions, their relations and an outlook to fields of research and open problems. Due to the amount of material presented, proofs are omitted or only hinted at, but a list of references at the end of each article makes it possible to access any further information needed.
The concise and clear style of the articles makes them pleasant reading, but the book’s real strength lies in the amount of material presented.
Let us pick some examples. In the article c-13: Generalized Continuities, we find a dozen notions of generalized continuity and their interrelations. When it comes to compactness, we find articles on Countable compactness, Pseudocompactness, Lindelöf spaces, Real compactness and, finally, on Generalizations of Paracompactness, and, there again, different notions are defined and discussed such as metacompactness, orthocompactness, sub-paracompactness and others. The book’s index covers 23 pages and enables one to quickly find what one is looking for.
Last but certainly not least it should be remarked that the authors of the individual articles are outstanding experts who have given and are still giving substantial contributions to their fields. The list of contributors reads like a Who is Who in General Topology. A beautiful book which can be highly recommended to anyone interested in topology.