General topology. 2nd corrected and expanded ed. (Allgemeine Topologie.) (German) Zbl 1320.54001

Bartsch, René

Berlin: De Gruyter (ISBN 978-3-11-040617-7/pbk; 978-3-11-040618-4/ebook). xvi, 306 p. (2015).

The book presents a very readable first introduction into the field of general topology, assuming only some basic knowledge of set-theory and real analysis at the start. In the second edition the author has added results about function spaces and two chapters on uniform spaces and hyperspaces.
The work now covers in eight chapters much of the basic classical material, besides a few topics related to function and hyperspaces, which are closer to the research interests of the author.
Chapter 1 discusses the set-theoretic foundations of the theory (for instance the axioms of set-theory, ordinals, cardinals, filters and ultrafilters). Chapter 2 then deals with the concept of a topological space (metric spaces, various methods to describe topologies and the notion of continuity). Chapter 3 treats the basic topological constructions (initial and final structures, trace and quotient topologies, products and coproducts). Separation axioms are discussed in Chapter 4 (for instance \(T_0\), \(T_1\), \(T_2\), \(T_3\), \(T_4\) spaces). Chapter 5 deals with the concept of compactness (its variants, \(T_{3\frac{1}{2}}\) spaces, function spaces, compactifications, paracompactness), while Chapter 6 is devoted to the concept of connectedness (including path connectedness, localisations). The theory of uniform spaces is discussed in Chapter 7 (uniformly continuous maps, completions, precompactness, uniformization). The final Chapter 8 contains a discussion of the idea of a hyperspace (Hausdorff metric, Bourbaki uniformity, Vietoris topology, hit-and-miss topologies, applications).
The book contains useful exercises that are spread throughout the text (together with helpful solutions and comments at the end of each chapter). The mathematical presentation of the material is reasonably demanding, without being pedantic. A common theme is the wish of the author to characterize introduced topological properties by (ultra)filter convergence. The theory is well illustrated by instructive examples. However the originality of the book mainly lies in the lively style in which it is written. The author regularly interrupts his sophisticated explanations with more or less appealing remarks and observations, which one normally does not find in a serious textbook of this kind. Let us mention a simple English example: It is stated on page 285 that “the topology…is not really hit-and-miss, because the miss-sets are missing”. The reviewer also noted that the word “daisy” even made it to the alphabetical index of the book, although mathematically this reference looks fairly unnecessary. In this way, even readers who care little about its challenging contents may find parts of the text entertaining and amusing.

Reviewer: Hans Peter Künzi (Rondebosch)

MSC:

54-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to general topology
54A05	Topological spaces and generalizations (closure spaces, etc.)
54Dxx	Fairly general properties of topological spaces

Keywords:

separation axioms; metric space; topological space; compactness; paracompactness; compactification; connectedness; uniform spaces; completion; hyperspaces; hit-and-miss topologies; function spaces; metrizability; filter; ultrafilter; convergence