Dimension and Extensions

J.M. Aarts, T. Nishiura

• Publisher: North Holland
• Year: 1993
• Print ISBN: 9780444897404
• eBook ISBN: 9780080887616
• Topic: Mathematics
• Subtopic: Geometry
• Index: Mathematics

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.

Table of contents

• Front Cover
• Dimension and Extensions
• Copyright Page
• Contents
• Preface
• Chapter I. The separable case in historical perspective
• Chapter II. Mappings into spheres
• Chapter III. Functions of inductive dimensional type
• Chapter IV. Functions of covering dimensional type
• Chapter V. Functions of basic dimensional type
• Chapter VI. Compactifications
• Chart 1. The absolute Borel classes
• Chart 2. Compactness dimension functions
• Bibliography
• List of symbols
• Index