Modern Dimension Theory (eBook)

Jun-iti Nagata (Autor)

N. G. De Bruijn, J. De Groot, A. C. Zaanen (Herausgeber)

eBook Download: PDF

2014 | 1. Auflage
268 Seiten
Elsevier Science (Verlag)
9781483275024 (ISBN)

Infos

Bibliotheca Mathematica, Volume 6: Modern Dimension Theory provides a brief account of dimension theory as it has been developed since 1941, including the principal results of the classical theory for separable metric spaces. This book discusses the decomposition theorem, Baire's zero-dimensional spaces, dimension of separable metric spaces, and characterization of dimension by a sequence of coverings. The imbedding of countable-dimensional spaces, sum theorem for strong inductive dimension, and cohomology group of a topological space are also elaborated. This text likewise covers the uniformly zero-dimensional mappings, theorems in euclidean space, transfinite inductive dimension, and dimension of non-metrizable spaces. This volume is recommended to students and specialists researching on dimension theory.

Inhalt

Front Cover 1
Modern Dimension Theory 4
Copyright Page 5
Table of Contents 8
PREFACE 6
CHAPTER I. INTRODUCTION 10
I. 1. Coverings 10
I. 2. Metrization 13
I. 3. Mappings 16
I. 4. Dimension 17
CHAPTER II. DIMENSION OF METRIC SPACES 20
II. 1. The lemmas to the sum theorem 20
II. 2. The sum theorem 23
II. 3. The decomposition theorem 27
II. 4. The product theorem 29
II. 5. The strong inductive dimension and the covering dimension 31
II. 6. Some theorems characterizing dimension 37
II. 7. Rank of coverings 43
II. 8. Normal families 53
CHAPTER III. MAPPINGS AND DIMENSION 58
III. 1. Stable value 58
III. 2. Extensions of mappings 61
III. 3. Essential mappings 64
III. 4. Continuous mappings which lower dimension 70
III. 5. Continuous mappings which raise dimension 77
III. 6. Baire's zero-dimensional spaces 82
III. 7. Uniformly zero-dimensional mappings 88
CHAPTER IV. DIMENSION OF SEPARABLE METRIC SPACES 99
IV. 1. Cantor manifolds 99
IV. 2. Dimension of En 102
IV. 3. Some theorems in Euclidean space 108
IV. 4. Imbedding 110
IV. 5. e-mappings 117
IV. 6. Dimension and measure 121
IV. 7. Dimension and the ring of continuous functions 126
CHAPTER V. DIMENSION AND METRIZATION 134
V. 1. Characterization of dimension by a sequence of coverings 135
V. 2. Length of coverings 142
V. 3. Dimension and metric function 146
V. 4. Another metric that characterizes dimension 163
CHAPTER VI. INFINITE-DIMENSIONAL SPACES 170
VI. 1. Countable-dimensional spaces 172
VI. 2. Imbedding of countable-dimensional spaces 179
VI. 3. Transfinite inductive dimension 186
VI. 4. General imbedding theorem 193
CHAPTER VII. DIMENSION OF NON-METRIZABLE SPACES 197
VII. 1. The general sum theorem 197
VII 2. Dimension of non-metrizable spaces 205
VII. 3. The sum theorem for strong inductive dimension 208
VII. 4. Dimension and mappings 217
CHAPTER VIII. DIMENSION AND COHOMOLOGY 229
VIII. 1. Homology group and cohomology group of a complex 229
Vin. 2. Cohomology group of a topological space 240
VIIL 3. Dimension and cohomology 243
VIII. 4. Dimension and homology 253
BIBLIOGRAPHY 258
Index 266

Produkt-Details

• Erscheint lt. Verlag: 12.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-13: 9781483275024 / 9781483275024