This volume is a translation of the first third of P. S. Aleksandrov’s Kombinatornaya Topologiya. An appendix on the analytic geometry of Euclidean n-space is also included. The volume, complete in itself, deals with certain classical problems such as the Jordan curve theorem and the classification of closed surfaces without using the formal techniques of homology theory. The elementary but rigorous treatment of these problems, the introductory chapters on complexes and coverings and their applications to dimension theory, and the large number of examples and pictures should provide an excellent intuitive background for further study in combinatorial topology.In Chapter I the references have been expanded to include a number of standard works in English. References to these and to the books and papers cited in Chapter I of the original are listed at the end of the chapter and correspond to the numbers enclosed in brackets in the body of the text. References in the remaining chapters are enclosed in brackets, capital letters referring to books and lower case letters to papers. These refer to the bibliography at the end of the book. The bibliography includes all papers mentioned in the original edition and a few which have been added by the translator.