Dimension and Extensions

Dimension and Extensions

1st Edition - January 28, 1993

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  • Authors: J.M. Aarts, T. Nishiura
  • eBook ISBN: 9780080887616

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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

  • 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.

Product details

  • No. of pages: 330
  • Language: English
  • Copyright: © North Holland 1993
  • Published: January 28, 1993
  • Imprint: North Holland
  • eBook ISBN: 9780080887616

About the Authors

J.M. Aarts

Affiliations and Expertise

Delft University of Technology, The Netherlands

T. Nishiura

Affiliations and Expertise

Wayne State University, Detroit, MI, USA

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