Theory of dimensions, finite and infinite

Title......Page 1
Table of Contents......Page 2
Preface......Page 4
1. Dimension theory of separable metric spaces......Page 6
1.1 Definition of the small inductive dimension......Page 7
1.2 The separation and enlargement theorems for dimension 0......Page 13
1.3 The sum, Cartesian product, universal space, compactification and embedding theorems for dimension 0......Page 20
1.4 Various kinds of disconnectedness......Page 29
1.5 The sum, decomposition, addition, enlargement, separation and Carthesian product theorems......Page 36
1.6 Definitions of the large inductive dimension and the covering dimension. Metric dimension......Page 45
1.7 The compactification and coincidence theorems. Characterization of dimension in terms of partitions......Page 52
1.8 Dimensional properties of Euclidean spaces and the Hilbert cube. Infinite dimensional spaces......Page 61
1.9 Characterization of dimension in terms of mappings to spheres. Cantor-manifolds. Cohomological dimension......Page 74
1.10 Characterization of dimension in terms of mappings to polyhedra......Page 84
1.11 The embedding and universal space theorems......Page 99
1.12 Dimension and mappings......Page 111
1.13 Dimension and inverse sequences of polyhedra......Page 119
1.14 Axioms for dimension......Page 128
2. The large inductive dimension......Page 132
2.1 Hereditarily normal and strongly hereditarily normal spaces......Page 132
2.2 Basic properties of the dimension Ind in normal and hereditarily normal spaces......Page 138
2.3 Basic properties of the dimension Ind in strongly hereditarily normal spaces......Page 149
2.4 Relations between the dimensions ind and Ind. Carthesian product theorems for the dimension Ind. Dimension Ind and mappings......Page 160
3. The covering dimension......Page 173
3.1 Basic properties of the dimension dim in normal spaces. Relations between the dimensions ind, Ind and dim.......Page 173
3.2 Characterizations of the dimension dim in normal spaces......Page 187
3.3 Dimension dim and mappings......Page 198
3.4 The compactification, universal space and Cartesian product theorems for the dimension dim. Dimension dim and inverse systems of compact spaces......Page 210
4. Dimension theory of metrizable spaces......Page 221
4.1 Basic properties of dimension in metrizable spaces......Page 222
4.2 Characterizations of dimension in metrizable spaces. The universal space theorems......Page 233
4.3 Dimension and mappings in metrizable spaces......Page 245
5. Countable-dimensional spaces......Page 257
5.1 Definitions and characterizations of countable-dimensional and strongly countable-dimensional spaces......Page 258
5.2 Basic properties of countable-dimensional and strongly countable-dimensional spaces......Page 266
5.3 The compactification and universal space theorems for countable-dimensional and strongly countable-dimensional spaces......Page 276
5.4 Countable dimensionality and mappings......Page 285
5.5 Locally finite-dimensional spaces......Page 293
6. Weakly infinite-dimensional spaces......Page 305
6.1 Definition and basic properties of weakly infinite-dimensional spaces......Page 305
6.2 An example of a totally disconnected strongly infinite-dimensional space......Page 317
6.3 Weak infinite dimensionality and mappings......Page 321
7. Transfinite dimensions......Page 330
7.1 Definitions and basic properties of the transfinite dimensions trind and trInd......Page 330
7.2 The separation, sum, enlargement, completion and universal space theorems for the transfinite dimensions trind and trInd......Page 343
7.3 Transfinite dimensions trker and trdim......Page 356
Bibliography......Page 370
List of special symbols......Page 398
Author index......Page 400
Subject index......Page 403