Elementary Topology A First Course Textbook in Problems O. Y. Viro, O. A. Ivanov, N. Y. Netsvetaev, V. M. Kharlamov Abstract. This book includes basic material on general topology, introduces algebraic topology via the fundamental group and covering spaces, and provides a background on topological and smooth manifolds. It is written mainly for students with a limited experience in mathematics, but determined to study the subject actively. The material is presented in a concise form, proofs are omitted. Theorems, however, are formulated in detail, and the reader is expected to treat them as problems. Foreword Genre, Contents and Style of the Book The core of the book is the material usually included in the Topology part of the two year Geometry lecture course at the Mathematical Department of St. Petersburg University. It was composed by Vladimir Abramovich Rokhlin in the sixties and has almost not changed since then. We believe this is the minimum topology that must be mastered by any student who has decided to become a mathematician. Students with research interests in topology and related fields will surely need to go beyond this book, but it may serve as a starting point. The book includes basic material on general topology, introduces algebraic topology via its most classical and elementary part, the theory of the fundamental group and covering spaces, and provides a background on topological and smooth manifolds. It is written mainly for students with a limited experience in mathematics, but who are determined to study the subject actively. The core material is presented in a concise form; proofs are omitted. Theorems, however, are formulated in detail. We present them as problems and expect the reader to treat them as problems. Most of the theorems are easy to find elsewhere with complete proofs. We believe that a serious attempt to prove a theorem must be the first reaction to its formulation. It should precede looking for a book where the theorem is proved. On the other hand, we want to emphasize the role of formulations. In the early stages of studying mathematics it is especially important to take each formulation seriously. We intentionally force a reader to think about each simple statement. We hope that this will make the book inconvenient for mere skimming. The core material is enhanced by many problems of various sorts and additional pieces of theory. Although they are closely related to the main material, they can be (and usually are) kept outside of the standard lecture course. These enhancements can be recognized by wider margins, as the next paragraph. The problems, which do not comprise separate topics and are intended exclusively to be exercises, are typeset with small face. Some of them are very easy and included just to provide additional examples. Few problems are difficult. They are to indicate relations with other parts of mathematics, show possible iii FOREWORD iv directions of development of the subject, or just satisfy an ambitious reader. Problems, whose solutions seem to be the most difficult (from the authors' viewpoint), are marked with a star, as in many other books. Further, we want to deliver additional pieces of theory (with respect to the core material) to more motivated and advanced students. Maybe, a mathematician, who does not work in the fields geometric in flavor, can afford the luxury not to know some of these things. Maybe, students studying topology can postpone this material to their graduate study. We would like to include this in graduate lecture courses. However, quite often it does not happen, because most of the topics of this sort are rather isolated from the contents of traditional graduate courses. They are important, but more related to the material of the very first topology course. In the book these topics are intertwined with the core material and exercises, but are distinguishable: they are typeset, like these lines, with large face, large margins, theorems and problems in them are numerated in a special manner described below. Exercises and illustrative problems to the additional topics are typeset with even wider margins and marked in a different way. Thus, the whole book contains four layers:ffl the core material,ffl exercises and illustrative problems to the core material,ffl additional topics,ffl exercises and illustrative problems to additional topics. The text of the core material is typeset with large face and smallest margins. The text of problems elaborating on the core material is typeset with small face and larger margins.