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Introduction 1 Eulers theorem
Surfaces g 4 Abstract spaces
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abelian barycentric subdivision base point boundary circle called choose closed sets closed subset closed surface coefficients combinatorial surface compact space connected space construction contains continuous function covering space curve cylinder defined definition denote dimension disc disjoint edge loop 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 Hq(K identification map identification space identification topology identity integer interior intersection inverse isomorphic join Klein bottle lemma limit point Mobius strip morphism neighbourhood nonempty one-one open cover open sets orbit space path-connected plane polygonal polyhedra polyhedron Problems projection proof of theorem prove real line real numbers result shown in Fig simplexes simplicial approximation simplicial complex simplicial map simply connected span sphere subcomplex subgroup subspace topology Suppose topological group topological space torus triangulable space union vertex vertices