Dimension and Extensions (eBook)

J.M. Aarts, T. Nishiura (Autoren)

eBook Download: PDF

1993 | 1. Auflage
330 Seiten
Elsevier Science (Verlag)
978-0-08-088761-6 (ISBN)

Infos

Two types of seemingly unrelated extension problems are discussed in this book. Their common focus is a long-standing problem of Johannes de Groot, the main conjecture of which was recently resolved. As is true of many important conjectures, a wide range of mathematical investigations had developed, which have been grouped into the two extension problems. The first concerns the extending of spaces, the second concerns extending the theory of dimension by replacing the empty space with other spaces.

The problem of de Groot concerned compactifications of spaces by means of an adjunction of a set of minimal dimension. This minimal dimension was called the compactness deficiency of a space. Early success in 1942 lead de Groot to invent a generalization of the dimension function, called the compactness degree of a space, with the hope that this function would internally characterize the compactness deficiency which is a topological invariant of a space that is externally defined by means of compact extensions of a space. From this, the two extension problems were spawned.

With the classical dimension theory as a model, the inductive, covering and basic aspects of the dimension functions are investigated in this volume, resulting in extensions of the sum, subspace and decomposition theorems and theorems about mappings into spheres. Presented are examples, counterexamples, open problems and solutions of the original and modified compactification problems.

Inhalt

Front Cover 1
Dimension and Extensions 4
Copyright Page 5
Contents 12
Preface 8
Chapter I. The separable case in historical perspective 16
I.1. A compactification problem 16
I.2. Dimensionsgrad 17
I.3. The small inductive dimension ind 18
I.4. The large inductive dimension Ind 24
I.5. The compactness degree de Groot’s problem
I.6. Splitting the compactification problem 35
I.7. The completeness degree 43
I.8. The covering dimension dim 56
I.9. The covering completeness degree 63
I.10. The s-compactness degree 68
I.11. Pol’s example 74
I.12. Kimura’s theorem 81
I.13. Guide to dimension theory 83
I.14. Historical comments and unsolved problems 84
Chapter II. Mappings into spheres 88
II.1. Classes and universe 89
II.2. Inductive dimension modulo a class P 91
II.3. Kernels and surplus 100
II.4. P-Ind and mappings into spheres 102
II.5. Covering dimensions modulo a class P 108
II.6. P-dim and mappings into spheres 115
II.7. Comparison of P-Ind and P-dim 121
II.8. Hulls and deficiency 124
II.9. Absolute Borel classes in metric spaces 127
II.10. Dimension modulo Borel classes 134
II.11. Historical comments and unsolved problems 141
Chapter III. Functions of inductive dimensional type 144
III.1. Additivity 145
III.2. Normal families 154
III.3. Optimal universe 161
III.4. Embedding theorems 175
III.5. Axioms for the dimension function 186
III.6. Historical comments and unsolved problems 196
Chapter IV. Functions of covering dimensional type 200
IV.1. Finite unions 201
IV.2. Normal families 207
IV.3. The Dowker universe D 212
IV.4. Dimension and mappings 219
IV.5. Historical comments and unsolved problems 230
Chapter V. Functions of basic dimensional type 232
V.1. The basic inductive dimension 232
V.2. Excision and extension 238
V.3. The order dimension 249
V.4. The mixed inductive dimension 256
V.5. Historical comments and unsolved problems 259
Chapter VI. Compactifications 262
VI.l. Wallman compactifications 263
VI.2. Dimension preserving compactifications 273
VI.3. The Fkeudenthal compactification 281
VI.4. The inequality K-Ind = K-Def 297
VI.5. Kimura’s characterization of K-def 302
VI.6. The inequality K-dim = K-def 315
VI.7. Historical comments and unsolved problems 327
Chart 1. The absolute Borel classes 87
Chart 2. Compactness dimension functions 261
Bibliography 330
List of symbols 342
Index 344

Produkt-Details

• Erscheint lt. Verlag: 28.1.1993
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 0-08-088761-9 / 0080887619
• ISBN-13: 978-0-08-088761-6 / 9780080887616