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
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
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
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
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