Introduction to Differential and Algebraic TopologyI︠U︡riĭ Grigorʹevich Borisovich Springer Netherlands, 31 mai 1995 - 492 pages Topology as a subject, in our opinion, plays a central role in university education. It is not really possible to design courses in differential geometry, mathematical analysis, differential equations, mechanics, functional analysis that correspond to the temporary state of these disciplines without involving topological concepts. Therefore, it is essential to acquaint students with topo logical research methods already in the first university courses. This textbook is one possible version of an introductory course in topo logy and elements of differential geometry, and it absolutely reflects both the authors' personal preferences and experience as lecturers and researchers. It deals with those areas of topology and geometry that are most closely related to fundamental courses in general mathematics. The educational material leaves a lecturer a free choice in designing his own course or his own seminar. We draw attention to a number of particularities in our book. The first chap ter, according to the authors' intention, should acquaint readers with topolo gical problems and concepts which arise from problems in geometry, analysis, and physics. Here, general topology (Ch. 2) is presented by introducing con structions, for example, related to the concept of quotient spaces, much earlier than various other notions of general topology thus making it possible for students to study important examples of manifolds (two-dimensional surfaces, projective spaces, orbit spaces, etc.) as topological spaces, immediately. |
Table des matières
Quotient space and quotient topology | 3 |
Product of topological spaces | 9 |
Category functor and algebraization of topological problems | 173 |
Droits d'auteur | |
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Expressions et termes fréquents
algebraic arbitrary atlas axiom boundary C-manifold called cell complex chain complexes chart commutative compact compact space concept connected Consider construct continuous mapping coordinates corresponding countable covering critical points defined definition denoted diffeomorphism dimension disc Dn+1 edge element equal Euclidean example Exercise exists fibre bundle finite number fixed point formula function ƒ functor fundamental group given gluing Hausdorff homeomorphism homology groups homology theory homotopy class homotopy equivalent induced intersection isomorphism lemma loop mapping f mapping ƒ matrix metric space Möbius strip n-dimensional obtain open neighbourhood open set orientation path path-connected plane polygon polyhedron preimage Proof properties Prove quotient space sequence Show simplex simplices simplicial complex simplicial mapping singular points smooth manifold sphere submanifold subset subspace t₁ tangent bundle tangent vector theorem topological space torus triangulation U₁ vector field vector space Verify vertices zero