The author is a specialist in analysis with a life long love for point set topology. As an academic teacher he prefers to go from the particular to the more general, so he starts in his first chapter with metric spaces. Then in a second chapter he develops general topological spaces, trying to take as much as possible from the main example, the metric spaces. In this chapter we find already the famous Tychonoff theorem about the product of compact spaces. The inevitable part of the proof which implies Zorn’s lemma is contained in a theorem of J. W. Alexander. To be informed about the separation axioms the impatient reader must wait until Chapter 3, where he is confronted with the “Agreement. All topological spaces encountered in this book will be assumed to be Hausdorff.” In regular spaces we are not interested. The theory of normal spaces is more deeply treated with the help of Dugundji’s book, which plays – and not only here – the role of a – let’s say – older half brother of the present treatment. Between regular spaces and normal spaces lives the distinguished class of completely regular spaces which is also well treated, with one exception: The author provides all tools to prove in a few lines the fact that this is exactly the class of subspaces of a compact space. However I could not find the explicit statement.
Once the reviewer asked P. S. Alexandroff: “What was the problem you gave Tychonoff for his dissertation, did you ask him if a product of compact spaces is compact?” – “Oh, no! I asked him to characterize subspaces of compact spaces.”
The book ends with a treatment of paracompact spaces. The author proves that every separable metric space is paracompact and mentions the fact that M. Rudin has a short proof for the assertion that all metric spaces are paracompact, apart from the proof of M. Stone in Dugundji’s book.
There is a relatively large collection of well investigated biographies which appear as footnotes, which are interesting and helpful, espacially for young readers. But where is Fréchet, who invented metric spaces, where is P. S. Alexandroff, the advisor of so many great names in the book (there is e.g., a chance to mention him underneath the one-point-compactification of locally compact spaces), why did he not include J. Dieudonné? Nevertheless the existence of these biographies can hardly be overestimated.
The book will be a success, a good introduction to point set topology and a valuable entrance for e.g., Dugundji’s book and other perhaps geometric treatments.