This book is a third edition following the 1968 edition (Elements of Modern Topology) and the 1988 edition (A Geometric Account of General Topology, Homotopy Types and the Fundamental Groupoid). The words of the author gives the general theme of the book: “a major emphasis of this and previous editions: the modelling of geometry, principally topology, by the algebra of groupoids”. A groupoid is a small category where every morphism is an isomorphism. It is a generalisation of groups regarded as a category having one object so that “groupoids can model more of the geometry than groups alone. This leads not only to more powerful theorems with simpler and more natural proofs but also to new theorems and new landscapes.” The book starts with basic topological notions (topology on reals, metric spaces, connectedness, compactness and constructions) and continues to objects of more geometric nature (cells, complexes, projective spaces and smash products). After an explanation of the basic categorical concepts, a fundamental groupoid is defined and studied. The next objects of explanation are cofibrations (with fibrations of groupoids), computation of the fundamental groupoid, covering spaces and covering groupoids, orbit spaces and orbit groupoids. At the end of the book, there are five pages of conclusions, an appendix (with basic facts about sets, functions and universal properties of products and sums), a glossary of some terms, a bibliography with almost 400 items and two indexes. Sections end with useful exercises and chapters with notes. It is no surprise that the book has reached its third edition.

Reviewer:

mihus