Dimension Theory (eBook)

Front Cover 1
Dimension Theory 4
Copyright Page 5
Contents 6
Preface 10
Acknowledgments 12
Chapter 1. Theory of Open Coverings 14
1. Complexes 15
2. Normality of Open Coverings 19
3. Extension of Coverings and Extensors 34
4. Identification Spaces and Absolute Retracts 41
5. Metrization Theorems 43
6. Inverse Limits and Compact Spaces 50
7. Totally Normal Spaces 55
Chapter 2. Dimension of Normal Spaces 57
8. Dimension Functions and 0-Dimensional Spaces 57
9. Covering Dimension of Normal Spaces 60
10. Large Inductive Dimension of Normal Spaces 69
11. Subset Theorem and Local Dimension 72
Chapter 3. Dimension of Metric Spaces 84
12. Fundamental Theorems for the Dimension of Metric Spaces 85
13. Topics on Nagata’s Metric and Infinite Dimensional Spaces 96
14. Sequences of Coverings Defining Dimension 103
15. Open Mappings and Dimension 107
Chapter 4. Gaps between Dimension Functions 112
16. A Compact Space Missing the Sum Theorem for ind and Ind 112
17. A Normal Space with ind = n Having Large dim 115
18. Compact Spaces with dim = n Having Large ind and Ind 120
19. A Normal Space with ind = 0, dim = 1, Ind = 2 127
Chapter 5. Dimension-Changing Closed Mappings 137
20. Relative Dimension 137
21. Dimension-Lowering Closed Mappings 142
22. Directed Families of Closed Coverings 143
23. Dimension-Raising Closed Mappings for Nonmetric Spaces 154
24. Dimension-Raising Closed Mappings for Metric Spaces 156
Chapter 6. Product Theorem and Expansion Theorem 161
25. Product Theorem for Ind 161
26. Product Theorem for dim 163
27. Expansion into Inverse Limits 167
Chapter 7. Metric-Dependent Dimension Functions 177
28. Dimension of the Complement of a Disjoint Sum 178
29. Relations between Metric-Dependent Dimension Functions 185
30. Spaces with d2 = [n/2] and dim > = n – 1
31. Spaces with m dim = [n/2] and dim > = n – 1
32. Spaces Illustrating the Dependence of d2, d3 and p dim on the Metric 198
33. A Space with d2 = 2, p dim = 3, dim = 4 203
Appendix: Cohomological Dimension Theory 209
34. Preliminaries: Cech Cohomology Group 209
35. Cohomological Dimension 212
36. A Characteristic Property of d(X: G ) 214
37. D(X:G) and d(X:G) 221
38. Sum Theorem 227
39. Bockstein’s Theorem I 232
40. Cantor Manifolds 238
41. Bockstein’s Theorem II 244
42. Products of Compact Metric Spaces and Paracompact Spaces 249
43. Subset Theorem and Product Theorem 254
References 258
Author Index 264
Subject Index 266
Pure and Applied Mathematics 271