Dimension and Extensions: Volume 48
North-Holland (Publisher)
Published on 28. January 1993
Book
Hardback
330 pages
978-0-444-89740-4 (ISBN)
Description
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.
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.
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.
Reviews / Votes
...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 NewsletterThis 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...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 NewsletterThis 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
More details
Series
Language
English
Place of publication
United States
Publishing group
Elsevier Science & Technology
Target group
College/higher education
Professional and scholarly
ISBN-13
978-0-444-89740-4 (9780444897404)
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Schweitzer Classification
Other editions
Additional editions

J. M. Aarts | T. Nishiura
Dimension and Extensions
E-Book
01/1993
North-Holland
€54.95
Available for download
Persons
Author
Delft University of Technology, The Netherlands
Wayne State University, Detroit, MI, USA
Content
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.