MR1969991 (Review)
Recent progress in general topology. II.
Edited by Miroslav Hu\v sek and Jan van Mill.
North-Holland, Amsterdam, 2002. xii+638 pp.
ISBN 0-444-50980-1
54-02 (54-06)
\{Vol. I has been reviewed [
MR 95g:54004].\}
Contents:
Alexander Arhangel\cprime skii [A. V. Arkhangel\cprime ski\u\i], Topological
invariants in algebraic environment (1--57) MR1969992;
Joni Baker and Kenneth Kunen, Matrices and ultrafilters (59--81) MR1969993;
Harold R. Bennett and David J. Lutzer, Recent developments in the topology
of ordered spaces (83--114) MR1969994;
Jan J. Dijkstra and Jan van Mill, Infinite-dimensional topology
(115--130) MR1969995;
Alan Dow, Recent results in set-theoretic topology (131--152) MR1969996;
Eli Glasner, Topics in topological dynamics, 1991 to 2001 (153--175) MR1969997;
Gilles Godefroy, Banach spaces of continuous functions on compact spaces
(177--199) MR1969998;
Gary Gruenhage, Metrizable spaces and generalizations (201--225) MR1969999;
Neil Hindman and Dona Strauss, Recent progress in the topological theory of
semigroups and the algebra of $\beta S$ (227--251) MR1970000;
\soft Lubica Holá and Jan Pelant, Recent progress in hyperspace topologies
(253--285) MR1970001;
Kazuhiro Kawamura, Some topics in geometric topology (287--311) MR1970002;
Hans-Peter A. Künzi, Quasi-uniform spaces in the year 2001 (313--344) MR1970003;
Witold Marciszewski, Function spaces (345--369) MR1970004;
Keye Martin, M. W. Mislove and G. M. Reed, Topology and domain theory
(371--394) MR1970005;
Roman Pol and Henryk Toru\'nczyk, Topics in dimension theory
(395--421) MR1970006;
Du\v san Repov\v s and Pavel V. Semenov, Continuous selections of
multivalued mappings (423--461) MR1970007;
Dmitri Shakhmatov [Dmitri\u\i B. Shakhmatov], Convergence in the presence of
algebraic structure (463--484) MR1970008;
S\l awomir Solecki, Descriptive set theory in topology (485--514) MR1970009;
Mikhail Tkachenko, Topological groups: between compactness and
$\aleph_0$-boundedness (515--543) MR1970010;
Essays (545--575) MR1970207;
A. V. Arhangel\cprime skii, H. R. Bennett, D. J. Lutzer, J. Dijkstra, J.\
van Mill, G. Godefroy, G. Gruenhage, N. Hindman, D. Strauss, K. Kawamura,
H.-P. Künzi, W. Marciszewski, K. Martin, M. W. Mislove, G. M. Reed, R.\
Pol, H. Toru\'nczyk, D. Repov\v s, P. V. Semenov, D. Shakhmatov, S.\
Solecki and M. Tkachenko, List of open problems and questions (577--597).
The predecessor to the book under review appeared in 1992
\ref[ Recent progress in general topology (Prague, 1991), North-Holland,
Amsterdam, 1992; MR 95g:54004]. That
book presented survey articles which described recent developments in
most of the primary subfields of general topology and its applications,
especially to algebra and analysis. Given the very positive reception
of that book by the topological research community, it is no surprise
that suggestions for an up-date by the publication of a second volume
ten years later have been met with the book under review. The
contributors are largely the group of invited speakers at the 2001
Prague Topological Symposium. The topics selected for this volume
reflect the change in focus in general topology in the last decade of
the twentieth century. Now applications to computer science are as
important as those to algebra and analysis. Room was not available for
every significant topic; for example, there is no chapter on continua
theory, but there is a very wide spread across the whole
field. Certainly there is much in this book for anyone with any sort
of topological interest. The table of contents is the best indicator
of the breadth of coverage. In addition to the nineteen chapters, each
a thorough discussion of the subfield under consideration, there is a
substantial list of open problems and questions. A fascinating new
feature of this volume is a set of short essays written by several
senior topologists, largely historical in character, but with room for
personal opinions and even predictions on where parts of the subject
might go from here. There are two very useful indexes---one for
authors and one for subjects. This is a book that every practising
topologist needs on their desk, and it is a must for all serious
mathematical research collections. The contributors and the editors
each deserve an accolade.
Reviewed by Ivan L. Reilly
