Topology I

A book by a Fields medal winner

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Introduction In the present essay, we attempt to convey some idea of the skeleton of topology, and of various topological concepts. It must be said at once that, apart from the necessary minimum, the subject-matter of this survey does not indude that subdiscipline known as "general topology" - the theory of general spaces and maps considered in the context of set theory and general category theory. (Doubtless this subject will be surveyed in detail by others. ) With this qualification, it may be daimed that the "topology" dealt with in the present survey is that mathematieal subject whieh in the late 19th century was called Analysis Situs, and at various later periods separated out into various subdisciplines: "Combinatorial topology", "Algebraic topology", "Differential (or smooth) topology", "Homotopy theory", "Geometrie topology". With the growth, over a long period of time, in applications of topology to other areas of mathematics, the following further subdisciplines crystallizedout: the global calculus of variations, global geometry, the topology of Lie groups and homogeneous spaces, the topology of complex manifolds and alge braic varieties, the qualitative (topologieal) theory of dynamical systems and foliations, the topology of elliptic and hyperbolic partial differential equations. Finally, in the 1970s and 80s, a whole complex of applications of topologie al methods was made to problems of modern physiesj in fact in several instances it would have been impossible to understand the essence of the real physical phenomena in question without the aid of concepts from topology.