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Sigma Series in Pure Mathematics -- Volume 2

   Enlarged Picture

Juniti Nagata

Modern Dimension Theory

294 pages, free electronic publication, ISBN 3-88538-002-1, 1983

The book gives a complete account of topological dimension theory.

"... worth reading. It gives a very good account of its subject, and its title is well deserved." (Bull. Amer. Math. Soc.)

Contents:

Contents (93 KB) v
Preface viii
Preface to the revised edition ix
     
  Chapter I: Introduction (293 KB)  
I.1 Coverings 1
I.2 Metrization 4
I.3 Mappings 7
I.4 Dimension 8
     
  Chapter II: Dimension of Metric Spaces (626 KB)  
II.1 Lemmas to sum theorem 10
II.2 Sum theorem 12
II.3 Decomposition theorem 16
II.4 Product theorem 17
II.5 Strong inductive dimension and covering dimension 20
II.6 Some theorems characterizing dimension 25
II.7 The rank of a covering 28
II.8 Normal families 31
     
  Chapter III: Mappings and Dimension (741 KB)  
III.1 Stable value 35
III.2 Extensions of mappings 37
III.3 Essential mappings 39
III.4 Some lemmas 43
III.5 Continuous mappings which lower dimension 46
III.6 Continuous mappings which raise dimension 49
III.7 Baire's zero-dimension spaces 51
III.8 Uniformly zero-dimensional mappings 55
     
  Chapter IV: Dimension of Separable Metric Spaces (863 KB)  
IV.1 Cantor manifolds 62
IV.2 Dimension of En 65
IV.3 Some theorems in Euclidean space 68
IV.4 Imbedding 69
IV.5 Epsilon-mappings 74
IV.6 Pontrjagin-Schnirelmann's theorem 76
IV.7 Dimension and measure 81
IV.8 Dimension and the ring of continuous functions 86
     
  Chapter V: Dimension and Metrization (675 KB)  
V.1 Characterization of dimension by a sequence of coverings 93
V.2 Length of coverings 98
V.3 Dimension and metric function 102
V.4 Another metric that characterizes dimension 114
     
  Chapter VI: Infinite-Dimensional Spaces (835 KB)  
VI.1 Countable-dimensional spaces 125
VI.2 Imbedding of countable-dimensional spaces 132
VI.3 Mappings and countable-dimensional spaces 139
VI.4 Transfinite inductive dimension 141
VI.5 Sum theorem for transfinite-inductive dimension 148
VI.6 General imbedding theorem 154
     
  Chapter VII: Dimension of Non-Metrizabel Spaces (1170 KB)  
VII.1 Sum theorem and subspace theorem for dim 158
VII.2 Dimensions of non-metrizable spaces 165
VII.3 Sum theorem and subspace theorem for Ind 170
VII.4 Characterization of dim by partitions 179
VII.5 Dimension and mappings 182
VII.6 Product theorem 192
VII.7 Characterization by Δk(X) 203
VII.8 Characterizations in terms of C(X) 207
     
  Chapter VIII: Dimension and Cohomology (841 KB)  
VIII.1 Homology group and cohomology group of a complex 213
VIII.2 Cohomology group of a topological space 225
VIII.3 Dimension and cohomology 228
VIII.4 Dimension and homology 239
     
  Bibliography (984 KB) 245
  Additional Bibliography 258
  List of Theorems 270
  List of Definitions 273
  Author index 275
  Subject index 281