An illustrated introduction to topology and homotopy
The book has two parts: The first one deals with general topology, starting with an introduction to set theory and the axiom of choice. The equivalence of the axiom of choice, Zorn's Lemma and the well-ordering theorem is mentioned but without proofs. Then the author does not proceed with general topological spaces but with metric spaces. In Chapter 3 general topological spaces are introduced. In Chapter 4 spaces with special properties, including CW complexes are discussed. Then he deals with product spaces, using two different topologies for infinitely many factors, preparing Tychonoff's theorem and applications. The other topology, the box topology for a product of spaces does not satisfy a Tychonoff theorem (i.e., infinite products of compact spaces need not be compact). In another chapter the separation axioms are treated, together with examples that a \(T_n\) space need not be a \(T_{n+\epsilon}\), \(\epsilon \in \{1/2,1\}\) space. All this enables him finally to deal with the Stone-Čech compactification and the determination of spaces which are subspaces of compact spaces (the completely regular spaces).NEWLINENEWLINEThe title of the second part (``Homotopy'') is somewhat misleading. Although the author introduces the concept of a homotopy, he deals only with the fundamental group; higher homotopy groups are only mentioned without giving any details. So this second part would have more properly come up under the title \textit{Fundamental group and combinatorial group theory}.NEWLINENEWLINEFor this and other reasons, this part appears as a new edition of Seifert-Threfall's famous book (which appeared before what is now called \textit{homotopy theory} was developed). Among other topics, one chapter deals with the Seifert-van Kampen theorem and its applications, and two chapters follow about covering spaces. This part contains objects like the horned sphere, the Borsuk-Ulam theorem (proved only for \(n=2\), like Brouwer's fixed point theorem) and it closes with group theoretical results like the assertion that subgroups of free groups are free and a result of Higman and the couple Neumann. The impression prevails that homotopy theory, in the way the author understands it, is first of all a remedy for proving group theoretical assertions.NEWLINENEWLINEIs this all that can be said about that book? No.NEWLINENEWLINEThere are ca. 500 pictures (called illustrations) which are mostly enlightening, some of them are not, however it is a new feature to combine abstract, advanced topology with so many illustrations and I think even the attempt is a success.NEWLINENEWLINEThere are many ``brief historical notes'' which are very helpful.NEWLINENEWLINEThe author loves examples, theorems, lemmas etc. and illustrations and he is on not so friendly terms with definitions, which are hidden in an ocean of text without a number to which you can legally refer. In one definition the author involves a previously given example ``\dots if the properties in example x are satisfied\dots''.NEWLINENEWLINEAt several occasions in the book concerning examples, counterexamples and illustrations I would say \textit{somewhat less would be somewhat more}. Nevertheless the book reflects interesting aspects and will find its readers.
scientific article; zbMATH DE number 6191822