Description
The book presents surveys describing recent developments in most of the primary subfields of
General Topology and its applications to Algebra and Analysis during the last decade. It follows freely
the previous edition (North Holland, 1992), Open Problems in Topology (North Holland, 1990) and
Handbook of Set-Theoretic Topology (North Holland, 1984). The book was prepared in
connection with the Prague Topological Symposium, held in 2001. During the last 10 years the focus
in General Topology changed and therefore the selection of topics differs slightly from those
chosen in 1992. The following areas experienced significant developments: Topological Groups, Function Spaces, Dimension Theory, Hyperspaces, Selections, Geometric Topology (including
Infinite-Dimensional Topology and the Geometry of Banach Spaces). Of course, not every important topic
could be included in this book.
Except surveys, the book contains several historical essays written by such eminent topologists as:
R.D. Anderson, W.W. Comfort, M. Henriksen, S. Mardeŝić, J. Nagata, M.E. Rudin, J.M. Smirnov (several reminiscences of L. Vietoris are added). In addition to extensive author and subject indexes, a list of all problems and questions posed in this book are added.
List of all authors of surveys:
A. Arhangel'skii, J. Baker and K. Kunen, H. Bennett and D. Lutzer, J. Dijkstra and J. van Mill, A. Dow, E. Glasner, G. Godefroy, G. Gruenhage, N. Hindman and D. Strauss, L. Hola and J. Pelant, K. Kawamura, H.-P. Kuenzi, W. Marciszewski, K. Martin and M. Mislove and M. Reed, R. Pol and H. Torunczyk, D. Repovs and P. Semenov, D. Shakhmatov, S. Solecki, M. Tkachenko.
Chapter
6. Topological groups and completions
7. Free topological groups
9. Duality theorems for topological groups
10. Some further results and problems on topological groups
Chapter 2. Matrices and ultrafilters
4. The Sikorski extension theorem
5. Hatsets in Stone spaces
Chapter 3. Recent developments in the topology of ordered spaces
3. Perfect ordered spaces
4. Base axioms related to metrizability
5. Diagonal and off-diagonal conditions in GO-spaces
6. Dugundji extension theory
7. Rudin's solution of Nikiel's problem, with applications to Hahn-Mazurkiewicz theory
8. Applications to Banach spaces
Chapter 4. Infinite-dimensional topology
2. Definitions and basic theory
3. Topological vector spaces
5. Homotopy dense imbeddings
6. Topological classification of semicontinuous functions
7. Hyperspaces of Peano continua
Chapter 5. Recent results in set-theoretical topology
3. Linearly Lindelöf spaces
4. Stone-(Čech compactification of N
5. Distributivity of N* × N*
6. Countable tightness in compact spaces
Chapter 6. Topics in topological dynamics, 1991 to 2001
2. Orbit equivalence of Cantor minimal dynamical systems
Chapter 7. Banach spaces of continuous functions on compact spaces
2. Linear classification of C(K) spaces
3. Renormings of C(K) spaces
4. Nonlinear classification of C(K) spaces
Chapter 8. Metrizable spaces and generalizations
2. Metrics, metrizable spaces, and mappings
5. Stratifiable and related spaces
6. Some higher cardinal generalizations
7. Moore and developable spaces
8. Bases with certain order properties
10. Sums of metrizable subspaces
Chapter 9. Recent progress in the topological theory of semigroups and the Algebra of βS
2. Topological and semitopological semigroups
3. Right (or left) topological semigroups
5. Applications to Ramsey Theory
Chapter 10. Recent progress in hyperspace topologies
2. Cardinal invariants of hyperspaces
4. Generalized metric properties of hyperspaces
5. Completeness properties of hyperspaces
6. Compactness in hyperspaces
Chapter 11. Some topics in geometric topology
2. Generalized manifolds and the recognition problem of topological manifolds
3. Cohomological dimension theory
4. Compactifications in geometric topology
Chapter 12. Quasi-uniform spaces in the year 2001
3. Extensions and completions
4. Functorial quasi-uniformities
5. Quasi-pseudometric spaces
6. Uniformizable ordered spaces
7. Hyperspaces and (multi)function spaces
Chapter 13. Function spaces
2. Function spaces on metrizable spaces
3. Function spaces on countable spaces
4. Products of function spaces
5. Condensations of function spaces
Chapter 14. Topology and domain theory
3. Models of topological spaces
Chapter 15. Topics in dimension theory
3. Weakly infinite-dimensional spaces and Haver's property C
5. Products of compact spaces
6. Products of non-compact spaces
7. Hereditarily indecomposable continua in dimension theory
8. Pushing compacta off affine manifolds in Euclidean spaces
10. Transfinite dimensions
11. The gap between the dimensions
12. Dimension-raising mappings with lifting properties
Chapter 16. Continuous selections of multivalued mappings
1. Solution of Michael's problem for C-domains
2. Selectors for hyperspaces
3. Relations between U- and L-theories
Chapter 17. Convergence in the presence of algebraic structure
1. Definitions of main convergence properties
2. Convergence properties in topological spaces
3. Convergence properties in topological groups
4. Convergence properties in groups with additional compactness conditions
5. Convergence properties in functions spaces Cp (X)
6. Convergence properties in products
7. Sequential order in topological groups and function spaces
Chapter 18. Descriptive set theory in topology
2. Polish topological group actions
3. Topologies on groups and ideals and complexity of their actions
4. Composants in indecomposable continua
5. Classifications of topological objects
Chapter 19. Topological groups: between compactness and No-boundedness
2. Countably compact groups
3. O-bounded and strictly o-bounded groups
1. The early development of infinite dimensional topology
2. Topological combinatorics: A peaceful pursuit
3. Topology related to rings of real-valued continuous functions. Where it has been and where it might be going
5. Looking back at modern general topology in the last century
6. Topology in the 20th Century
8. Reminiscences of L. Vietoris
Chapter 21. List of open problems and questions
2. Bennett, H.R. and D.J. Lutzer
3. Dijkstra, J. and J. van Mill
6. Hindman, N. and D. Strauss
10. Martin, K., M.W. Mislove and G.M. Reed
11. Pol, R. and H. Toruficzyk
12. Repovš, D. and P.V. Semenov
Recent Progress in General Topology II
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