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Sigma Series in Pure Mathematics -- Volume 4

   Enlarged Picture

R. Engelking, K. Sieklucki

Topology. A Geometric Approach
Translated from the Polish by A. Ostaszewski

viii + 430 pages, hard cover, ISBN 3-88538-004-8, EUR 75.00, 1992

This is an introduction to algebraic, geometric and general topology. Starting from elementary concepts, the theories of metric spaces, polyhedra, complexes, homotopy, and the fundamental group are developed. Then follow chapters on the topology of Euclidean spaces and topological manifolds. Finally, complete metric spaces, continua, and dimension are discussed.


Zentralblatt-Review


Contents:

Foreword vii
     
  Chapter 0: Introduction  
0.1 Set theory 1
0.2 Algebra 4
0.3 Analysis 6
0.4 Geometry 7
     
  Chapter 1: Metric spaces  
1.1 Concept of a metric space 14
1.2 Operations on metric spaces 21
1.3 Maps on metric spaces 21
1.4 Metric concepts 32
1.5 Convergence and limits 36
1.6 Open and closed sets 41
1.7 Connected spaces 51
1.8 Compact spaces 59
1.9 Complete spaces 65
1.10 Metric and topological concepts in Euclidean spaces 68
1.S Supplements 76
1.P Problems 82
     
  Chapter 2: Polyhedra  
2.1 Simplices 87
2.2 Simplicial complexes 91
2.3 Polyhedra 94
2.4 Subdivisions 99
2.5 Simplicial maps 102
2.6 Cell complexes 107
2.S Supplements 112
2.P Problems 117
     
  Chapter 3: Homotopy  
3.1 Extensions of continuous maps 122
3.2 Homotopic maps 131
3.3 Fibrations and coverings 140
3.4 The fundamental group 150
3.S Supplements 169
3.P Problems 173
     
  Chapter 4: The topology of Euclidean spaces  
4.1 Maps into spheres 177
4.2 Topological invariance of certain properties of sets 182
4.3 The theory of position 186
4.4 Various examples 198
4.S Supplements 206
4.P Problems 208
     
  Chapter 5: Manifolds  
5.1 The concept of a topological manifold 211
5.2 Orientability of a manifold 219
5.3 Pastings and cuttings 222
5.4 Classification of 1- and 2-dimensional manifolds 228
5.S Supplements 240
5.P Problems 243
     
  Chapter 6: Metric spaces II  
6.1 Countable products of metric spaces 247
6.2 Spaces of maps 254
6.3 Separable spaces 259
6.4 Complete spaces and completions 266
6.5 Continua 276
6.6 Absolute retracts and absolute neighbourhood retracts 287
6.7 The dimension of separable metric spaces 295
6.8 Dimension in Euclidean spaces 304
6.S Supplements 312
6.P Problems 322
     
  Chapter 7: Topological spaces  
7.1 The concept of a topological space 332
7.2 Maps on topological spaces 342
7.3 Separation axioms 348
7.4 Operations on topological spaces 356
7.5 Compact spaces and compactifications 373
7.6 Metrization of topological spaces. Paracompact spaces 391
7.S Supplements 397
7.P Problems 406
     
  Bibliography 418
     
  Subject index 419