Book review

In this increasingly more interesting book-series, as interesting incidentally as the companion Sigma Series in Applied Mathematics, Heldermann Verlag served the mathematical community well by publishing a new edition in 1989 of Engelking's acclaimed General Topology, of which the previous 1977 edition had long ago sold out. This series is now enriched with the publication of a new book by Engelking and Sieklucki which gives an introduction to geometric and general topology (as well as a small part of algebraic topology) using a minimum of prerequisites to obtain maximum results.

Topology, a Geometric Approach has many of its stylistic features in common with General Topology: few words are wasted, following opening remarks and definitions at the start of sections, practically every paragraph is either a theorem, assertion, lemma, corrolary or example; each chapter ends with supplementary notes (bibliographical and historical remarks, and more extensive discussions of additional, more advanced, theory); the exercises come in two guises, first the, mostly easy but sometimes difficult, 'exercises' that test command of the theory, and the 'problems', which can be very difficult, that ask for proofs of results that supplement the main text (and are often provided with hints).

The prerequisites, elementary set theory, elementary group theory and geometry and some basic results from analysis, will generally be learned in the first year of studying mathematics. To refresh memory, the basic terminology needed from the prerequisites is compactly but adequately introduced in Chapter 0. Chapter 1 treats the elementary concepts from the topology of metric spaces to set the stage for the book proper that starts in Chapter 2 with an introduction to polyhedra (simplices, simplicial complexes and maps, and cell complexes). Polyhedra have the advantage of appealing to the intuition of the reader (certainly with the profuse illustrations) while at the same time serving as paradigms for many more abstract concepts and spaces that will follow. In Chapter 3, the basic notions from homotopy theory are developed (extensions of continuous maps, homotopy, fibrations and coverings, and the fundamental group). Chapters 4 and 5 discuss the topology of Euclidian spaces and topological manifolds (with particular attention to surfaces). Chapter 6 continuous the discussion of metric spaces from Chapter 1 with separable spaces, complete spaces and continua, and sections on absolute retracts and dimension theory. Finally, Chapter 7 contains the elements of abstract topological space theory (including compactifications and metrisability).

Of course there is overlap between the contents of this book and the first author's General Topology and Dimension Theory. However, the present book is not intended to supplement either of the others and has a different audience in mind. It can be highly recommended as an excellent introduction to topology, without much prior knowledge, and can be sure to stimulate geometric reasoning in the way the student will think about the abstract general topological concepts. The extensive use, in more than 160 figures, of graphical metaphores to illustrate key concepts and constructions and the graphical representations of some of the proofs, are particularly conducive to this effect. Topology, a Geometric Approach is a valuable contribution to the (short) list of good introductions to topology that are in use today, and it deserves to be widely read. On my, highly subjective, scale of 1 to 5, I give this book 4 stars.