Dimension and Extensions book cover

Dimension and Extensions

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.

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Published: January 1993

Imprint: North-holland

ISBN: 978-0-444-89740-4

Reviews

  • ...an excellent, complete survey... A systematic treatment, both of the properties of dimension-like functions as well as of the classes of extensions, a kind attitude to the reader and perfect organization are the main highlights of this fine book.
    European Mathematical Society Newsletter


    This excellently written, exciting book is a portrait of a living and dynamic area ..... It should be required reading for anyone interested in dimension theory .... The exposition is masterful.
    Bulletin of the American Mathematical Society

Contents

  • The Separable Case in Historical Perspective. Mappings into Spheres. Functions of Inductive Dimensional Type. Functions of Covering Dimensional Type. Functions of Basic Dimensional Type. Compactifications. Charts: The Absolute Borel Classes. Compactness Dimension Functions. Bibliography. Index.

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