Preface Modern topology uses many different methods. In this book, we largely investigate the methods of combinatorial topology and the methods of dif- ferential topology the former reduce studying topological spaces to investi- gation of their partitions into elementary sets, such as simplices, or covers by some simple sets, while the latter deal with smooth manifolds and smooth maps. Many topological problems can be solved by using any of the two approaches, combinatorial or differential in such cases, we discuss both of them. Topology has its historical origins in the work of Riemann Riemann's investigation was continued by Betti and Poincaxe. While studying mul- tivalued analytic functions of a complex variable, Riemann realized that, rather than in the plane, multivalued functions should be considered on two-dimensional surfaces on which they are single-valued. In these con- siderations, two-dimensional surfaces arise by themselves and are defined intrinsically, independently of their particular embeddings in E3 they are obtained by gluing together overlapping plane domains. Then, Riemann introduced the notion of what is known as a (multidimensional) manifold (in the German literature, Riemann's term Mannigfaltigkeit is used). A manifold of dimension n, or n-manifold, is obtained by gluing together over- lapping domains of the space Mn. Later, it was recognized that to describe continuous maps of manifolds, it suffices to know only the structure of the open subsets of these manifolds. This was one of the most important rea- sons for introducing the notion of topological space this is a set endowed with a topology, that is, a system of subsets (called open sets) with certain properties. vn
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