In this broad introduction to topology, the author searches for topological invariants of spaces, together with techniques for calculating them. Students with knowledge of real analysis, elementary group theory, and linear algebra will quickly become familiar with a wide variety of techniques and applications involving point-set, geometric, and algebraic topology. Over 139 illustrations and more than 350 problems of various difficulties will help students gain a rounded understanding of the subject.
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A spacefilling curve
Compactness and connectedness
The identification topology
The fundamental group
Degree and Lefschetz number
Knots and covering spaces
Generators and relations
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abelian group barycentric subdivision base point boundary circle called chain map choose closed surface coefficients combinatorial surface construction contains continuous function covering space curve cylinder defined definition denote dimension disc disjoint edge loop edge path element equivalent euclidean space Euler characteristic example Figure finite number fixed points free group fundamental group give given Hausdorff space homeo homeomorphism homology groups homotopy type identity induced infinite cyclic integer interior intersection isomorphic join Kampen's theorem Klein bottle knot group lemma limit point map f Möbius strip morphism neighbourhood nonempty one–one open cover open sets orbit space oriented q-simplex path-connected polygonal polyhedra polyhedron Problems proof of theorem prove q-cycle real line real numbers result shown in Fig simplexes simplicial approximation simplicial complex simplicial map simply connected sphere subcomplex subgroup Suppose thicken topological group topological space torus triangulable space union vertex vertices