Modern Dimension Theory

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.

Chapter I. Introduction

I.1 Coverings

I.2. Metrization

I.3. Mappings

I.4. Dimension

Chapter II. Dimension of Metric Spaces

II.1. The Lemmas to the Sum Theorem

II.2. The Sum Theorem

II.3. The Decomposition Theorem

II.4. The Product Theorem

II.5. The Strong Inductive Dimension and the Covering Dimension

II.6. Some Theorems Characterizing Dimension

II.7. Rank of Coverings

II.8. Normal Families

Chapter III. Mappings and Dimension

III.1. Stable Value

III.2. Extensions of Mappings

III.3. Essential Mappings

III.4. Continuous Mappings which Lower Dimension

III.5. Continuous Mappings which Raise Dimension

III.6. Baire's Zero-Dimensional Spaces

III.7. Uniformly Zero-Dimensional Mappings

Chapter IV. Dimension of Separable Metric Spaces

IV.1. Cantor Manifolds

IV.2. Dimension of En

IV.3. Some Theorems in Euclidean Space

IV.4. Imbedding

IV.5. Є-mappings

IV.6. Dimension and Measure

IV.7. Dimension and the Ring of Continuous Functions

Chapter V. Dimension and Metrization

V.l. Characterization of Dimension by a Sequence of Coverings

V.2. Length of Coverings

V.3. Dimension and Metric Function

V.4. Another Metric that Characterizes Dimension

Chapter VI. Infinite-Dimensional Spaces

VI.1. Countable-Dimensional Spaces

VI.2. Imbedding of Countable-Dimensional Spaces

VI.3. Transfinite Inductive Dimension

VI.4. General Imbedding Theorem

Chapter VII. Dimension of Non-Metrizable Spaces

VII.1. The General Sum Theorem

VII.2. Dimension of Non-Metrizable Spaces

VII.3. The Sum theorem for Strong Inductive Dimension

VII.4. Dimension and Mappings

Chapter VIII. Dimension and Cohomology

VIII.1. Homology Group and Cohomology Group of a Complex

VIII.2. Cohomology Group of a Topological Space

VIII.3. Dimension and Cohomology

VIII.4. Dimension and Homology

Bibliography

Index