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**Topology.
2nd ed.**
*(English)*
Zbl 0951.54001

Upper Saddle River, NJ: Prentice Hall. xvi, 537 p. (2000).

The present book is an expanded and rewritten edition of an earlier book by the author [Topology. A first course. (1975; Zbl 0306.54001)]. As in the first edition, the text material is divided into two parts: (Part I) General Topology and (Part II) Algebraic Topology. Some of the chapters from Part II of the first edition have been moved with no substantial changes to Part I of the second edition. There is a new chapter in Part I giving an introduction to Baire spaces and dimension theory. The final chapter in Part II of the previous edition, which dealt with algebraic topology, has been substantially expanded, and it has become Part II of the new edition. This part of the book treats with some thoroughness the notions of fundamental group and covering space, along with their many and varied applications, in particular, to the classification of surfaces and covering spaces. The topics covered are a very good selection and give to the reader a good introduction to algebraic topology. In the present edition exercises have been reworked and to some chapters new supplementary exercises have been attached.

The chapter titles are: (Part I) 1. Set theory and logic; 2. Topological spaces and continuous functions; 3. Connectedness and compactness; 4. Countability and separation axioms; 5. The Tychonoff theorem; 6. Metrization theorems and paracompactness; 7. Complete metric spaces and function spaces; 8. Baire spaces and dimension theory; (Part II) 9. The fundamental group; 10. Separation theorems in the plane; 11. The Seifert-van Kampen theorem; 12. Classification of surfaces; 13. Classification of covering spaces; 14. Applications to group theory.

The book is intended as an introductory course in general and algebraic topology at the senior undergraduate or first-year graduate level. The text is well written and it could be used as a one- or two-semester course by selecting from among the topics covered in the book, e.g., according to hints given by the author in the preface. There are many excellent illustrations.

The chapter titles are: (Part I) 1. Set theory and logic; 2. Topological spaces and continuous functions; 3. Connectedness and compactness; 4. Countability and separation axioms; 5. The Tychonoff theorem; 6. Metrization theorems and paracompactness; 7. Complete metric spaces and function spaces; 8. Baire spaces and dimension theory; (Part II) 9. The fundamental group; 10. Separation theorems in the plane; 11. The Seifert-van Kampen theorem; 12. Classification of surfaces; 13. Classification of covering spaces; 14. Applications to group theory.

The book is intended as an introductory course in general and algebraic topology at the senior undergraduate or first-year graduate level. The text is well written and it could be used as a one- or two-semester course by selecting from among the topics covered in the book, e.g., according to hints given by the author in the preface. There are many excellent illustrations.

Reviewer: Z.Karno (Białystok)

### MSC:

54-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to general topology |

55-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic topology |