Basic TopologyIn 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. |
Contents
| 1 | |
| 4 | |
Surfaces | 8 |
Abstract spaces | 12 |
A classification theorem | 16 |
Topological invariants | 19 |
Continuity 1 Open and closed sets | 27 |
Continuous functions | 32 |
Topological groups | 73 |
Orbit spaces | 78 |
The fundamental group 1 Homotopic maps | 87 |
Construction of the fundamental group | 92 |
Calculations | 96 |
Homotopy type | 103 |
The Brouwer fixedpoint theorem | 110 |
Separation of the plane | 112 |
A spacefilling curve | 36 |
The Tietze extension theorem | 38 |
Compactness and connectedness 1 Closed bounded subsets of E | 43 |
The HeineBorel theorem | 44 |
Properties of compact spaces | 47 |
Product spaces | 51 |
Connectedness | 56 |
Joining points by paths | 61 |
Identification spaces 1 Constructing a Möbius strip | 65 |
The identification topology | 66 |
The boundary of a surface | 115 |
Triangulations | 119 |
Triangulating orbit spaces | 140 |
Triangulation and orientation | 153 |
Simplicial homology | 173 |
Invariance | 188 |
The BorsukUlam theorem | 202 |
The knot group | 216 |
The Alexander polynomial | 234 |
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Common terms and phrases
abelian antipodal base point boundary circle called choose closed sets closed surface coefficients combinatorial surface compact space construction contains continuous function covering space curve cylinder D₁ defined definition denote dimension disc disjoint edge loop element equivalent euclidean space Euler characteristic example Figure finite number fundamental group give given H₄(K Hausdorff space homeo homeomorphism homology groups homotopy type identification map identification space identification topology identity induced integer interior intersection isomorphic Klein bottle lemma Let f limit point map f Möbius strip morphism neighbourhood nonempty one-one open cover open sets orbit space path-connected plane polygonal polyhedron Problems proof of theorem prove real line real numbers result shown in Fig simplexes simplicial approximation simplicial complex simplicial map simply connected sphere subcomplex subgroup subspace topology Suppose topological group topological space torus union v₁ vertex vertices X₁ π₁