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| THE INFINITE-DIMENSIONAL TOPOLOGY OF FUNCTION SPACES, 64
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By
J. van Mill, Vrije Universiteit, Department of Mathematics and Computer Science, Amsterdam, The Netherlands
Included in series
North-Holland Mathematical Library, 64
Description
In this book we study function spaces of low Borel complexity.
Techniques from general topology, infinite-dimensional topology, functional
analysis and descriptive set theory
are primarily used for the study of these spaces. The mix of
methods from several disciplines makes
the subject
particularly interesting. Among other things, a complete and self-contained proof of the Dobrowolski-Marciszewski-Mogilski
Theorem that all function spaces of low Borel complexity are topologically homeomorphic, is presented.
In order to understand what
is going on, a solid background in
infinite-dimensional topology is needed. And for that a fair amount of knowledge of dimension theory
as well as ANR theory is needed. The necessary material was partially covered in our previous book `Infinite-dimensional topology, prerequisites
and introduction'. A selection of what was done there can be found here as well, but completely revised and at many places expanded with
recent results. A `scenic' route has been chosen towards the
Dobrowolski-Marciszewski-Mogilski Theorem, linking the
results needed for
its proof to interesting recent research developments in dimension theory and infinite-dimensional topology.
The first five chapters
of this book are intended as a text for
graduate courses in topology. For a course in dimension theory, Chapters 2 and 3 and part of
Chapter 1 should be covered. For a course in infinite-dimensional topology, Chapters 1, 4 and 5. In Chapter 6, which deals with function
spaces, recent research results are discussed. It could also be used for a graduate course in topology but its flavor is more that of
a research monograph than of a textbook; it is therefore
more suitable as a text for a research seminar. The book
consequently has the
character of both textbook and a research monograph. In Chapters 1 through 5, unless stated
otherwise, all spaces under discussion are
separable and
metrizable. In Chapter 6 results for more general classes of spaces are presented.
In Appendix A for easy reference
and some basic facts that are important in the book have been collected. The book is not intended as a basis for a course in topology;
its purpose is to collect knowledge about general topology.
The exercises in the book serve three purposes: 1) to test the reader's
understanding of the material 2) to supply proofs of statements that are used in the text, but are not proven there
3) to provide additional
information not covered by the text.
Solutions to selected exercises have been included in Appendix B.
These exercises are important
or difficult.
Contents
Introduction.
Chapter 1. Basic topology. Chapter 2. Basic combinatorial topology. Chapter 3. Basic dimension theory. Chapter 4. Basic
ANR theory. Chapter 5. Basic infinite-dimensional topology. Chapter 6. Function spaces. Appendix A. Preliminaries. Appendix B. Answers
to selected exercises. Appendix C. Notes and comments. Bibliography. Special Symbols. Author Index. Subject Index.
| Bibliographic details |
Hardbound, 644 pages, publication date: JUN-2001
ISBN-13: 978-0-444-50557-6
ISBN-10: 0-444-50557-1
Imprint: NORTH-HOLLAND
Paperback, 642 pages, publication date: MAY-2002
ISBN-13: 978-0-444-50849-2
ISBN-10: 0-444-50849-X
Imprint: NORTH-HOLLAND
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Last update: 4 Sep 2009
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