Topology

Contents......Page 8
Preface......Page 4
A Note on Set-Theoretic Concepts......Page 6
1-1 Introduction......Page 12
1-2 Topological spaces......Page 16
1-3 Basis and subbasis of a topology......Page 17
1-4 Metric spaces and metric topologies......Page 20
1-5 Continuous mappings......Page 23
1-6 Connectedness. Subspace topologies......Page 25
1-7 Compactness......Page 29
1-8 Product spaces......Page 32
1-9 Some theorems in logic......Page 34
1-10 The Tychonoff theorem......Page 36
1-11 Function spaces......Page 39
1-12 Uniform continuity and uniform spaces......Page 41
*1-13 Kuratowski's closure operation......Page 43
1-14 Topological groups......Page 44
2-2 Separation axioms......Page 48
2-3 T_3- and T_4-spaces......Page 51
2-4 Continua in Hausdorff spaces......Page 54
2-5 The interval and the circle......Page 63
2-6 Real functions on a space......Page 67
2-7 The Tietze extension theorem......Page 70
2-8 Completely separable spaces......Page 75
2-9 Mappings into Hilbert space. A metrization theorem......Page 78
2-10 Locally compact spaces......Page 82
*2-11 Paracompact spaces......Page 88
*2-12 A general metrization theorem......Page 91
2-13 Complete metric spaces. The Baire-Moore theorem......Page 92
2-14 Inverse limit systems......Page 102
*2-15 A characterization of the Cantor set......Page 108
2-16 Limits inferior and superior......Page 111
3-1 Locally connected spaces......Page 116
3-2 Arcs, arcwise connectivity, and accessibility......Page 126
3-3 Mappings of the interval......Page 133
3-4 Mappings of the Cantor set......Page 137
3-5 The Hahn-Mazurkiewicz theorem......Page 140
3-6 Decomposition spaces and continuous transformations......Page 143
3-7 Monotone and light mappings......Page 147
*3-8 Indecomposable continua......Page 150
*3-9 Dimension theory......Page 156
4-1 Introduction......Page 160
4-2 Homotopic mappings......Page 161
4-3 Essential and inessential mappings......Page 165
4-4 Homotopically equivalent spaces......Page 168
4-5 The fundamental group......Page 170
4-6 Knots and related imbedding problems......Page 185
4-7 The higher homotopy groups......Page 189
4-8 Covering spaces......Page 199
*4-9 Homotopy connectedness and homotopy local connectedness......Page 201
5-2 Vector spaces......Page 204
5-3 E^n as a vector space over E^l. Barycentric coordinates......Page 206
5-4 Geometric complexes and polytopes......Page 210
5-5 Barycentric subdivision......Page 217
5-6 Simplicial mappings and the simplicial approximation theorem......Page 220
5-7 Abstract simplicial complexes......Page 224
*5-8 An imbedding theorem for polytopes......Page 225
6-1 Introduction......Page 229
6-2 Oriented complexes......Page 233
6-3 Incidence numbers......Page 234
6-4 Chains, cycles, and groups......Page 236
6-5 The decomposition theorem for abelian groups. Betti numbers and torsion coefficients......Page 245
6-6 Zero-dimensional homology groups......Page 249
6-7 The Euler-Poincaré formula......Page 252
6-8 Some general remarks......Page 254
*6-9 Universal coefficients......Page 255
6-10 Simplicial mappings again......Page 259
6-11 Chain-mappings......Page 264
6-12 Cone-complexes......Page 268
6-13 Barycentric subdivision again......Page 269
6-14 The Brouwer degree......Page 274
6-15 The fundamental theorem of algebra, an existence proof......Page 280
6-16 The no-retraction theorem and the Brouwer fixed-point theorem......Page 282
6-17 Mappings into spheres......Page 284
7-1 Relative homology groups......Page 293
7-2 The exact homology sequence......Page 295
7-3 Homomorphisms of exact sequences......Page 298
7-4 The excision theorem......Page 299
7-5 The Mayer-Vietoris sequence......Page 301
7-7 The Eilenberg-Steenrod axioms for homology theory......Page 304
7-8 Relative homotopy theory......Page 306
7-9 Cohomology groups......Page 308
7-10 Relations between chain and cochain groups......Page 311
7-11 Simplicial and chain-mappings......Page 314
7-12 The cohomology product......Page 317
7-13 The cap-product......Page 321
7-14 Relative cohomology theory......Page 323
7-15 Exact sequences in cohomology theory......Page 326
7-16 Relations between homology and cohomology groups......Page 328
8-1 Cech Homology Theory (introduction)......Page 331
8-2 The topological invariance of simplicial homology groups......Page 343
8-3 Cech homology theory (continued)......Page 348
8-4 Induced homomorphisms......Page 350
*8-5 Singular homology theory......Page 352
*8-6 Vietoris homology theory......Page 357
*8-7 Homology local connectedness......Page 358
8-8 Some topology of the n-sphere......Page 361
Books......Page 376
Papers, etc......Page 377
Index......Page 382