The InfiniteDimensional Topology of Function Spaces
Author: 
J. van Mill 
Publisher: 

Year: 
2001 
Series: 
NorthHolland Mathematical Library, 64

Review:


Description
In this book we study function spaces of low Borel
complexity. Techniques from general topology, infinitedimensional 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
selfcontained proof of the DobrowolskiMarciszewskiMogilski 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 infinitedimensional 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 `Infinitedimensional
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 DobrowolskiMarciszewskiMogilski Theorem, linking the results needed
for its proof to interesting recent research developments in dimension
theory and infinitedimensional 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 infinitedimensional topology,
Chapters 1, 4 and 5 should be covered. 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. This appendix is not
intended
as a basis for a course in topology; its purpose is to collect
necessary 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 infinitedimensional 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.