Recent Progress in General Topology II

by Husek; van Mill

• ISBN13: 9780444509802
• ISBN10: 0444509801
• Format: Hardcover
• Copyright: 2002-11-13
• Publisher: Elsevier Science

Summary

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. Mardescirc;icacute;, 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.

Table of Contents

Topological invariants in algebraic environment	1	(59)
A.V. Arhangel'skii
Introduction	3	(1)
Topologies on groups	3	(3)
Some relations between invariants of semitopological and paratopological groups	6	(3)
Some special algebraic structures and topologies	9	(5)
Extremal topologies and various algebraic structures	14	(7)
Topological groups and completions	21	(10)
Free topological groups	31	(6)
The Bohr topology	37	(2)
Duality theorems for topological groups	39	(1)
Some further results and problems on topological groups	40	(19)
References	48	(11)
Matrices and ultrafilters	59	(24)
J. Baker
K. Kunen
Introduction	61	(4)
Hatpoints and hatsets	65	(3)
Matrices	68	(4)
The Sikorski extension theorem	72	(2)
Hatsets in Stone spaces	74	(5)
Avoiding P-points	79	(4)
References	80	(3)
Recent developments in the topology of ordered spaces	83	(32)
H.R. Bennett
D.J. Lutzer
Introduction	85	(1)
Orderability	85	(1)
Perfect ordered spaces	86	(5)
Base axioms related to matrizability	91	(6)
Diagonal and off-diagonal conditions in GO-spaces	97	(6)
Dugundji extension theory	103	(1)
Rudin's solution of Nikiel's problem, with applications to Hahn-Mazurkiewicz theory	104	(1)
Applications to Banach spaces	105	(1)
Products of Go-spaces	106	(9)
References	109	(6)
Infinite-dimensional topology	115	(16)
J.J. Dijkstra
J. van Mill
Introduction	117	(1)
Definitions and basic theory	117	(1)
Topological vector spaces	118	(4)
Functions spaces	122	(1)
Homotopy dense imbeddings	123	(2)
Topological classification of semicontinuous functions	125	(2)
Hyperspaces of Peano continua	127	(4)
References	128	(3)
Recent results in set-theoretical topology	131	(22)
A. Dow
Introduction	133	(1)
Standard tools	133	(2)
Linearly Lindelof spaces	135	(1)
Stone-Cech compactification of N	136	(3)
Distributivity of N* x N*	139	(2)
Countable tightness in compact spaces	141	(12)
References	150	(3)
Topics in topological dynamics, 1991 to 2001	153	(24)
E. Glasner
Introduction	155	(1)
Orbit equivalence of Cantor minimal dynamical systems	155	(5)
Williams' conjecture	160	(8)
Mean dimension	168	(9)
References	173	(4)
Banach spaces of continuous functions on compact spaces	177	(24)
G. Godefroy
Introduction	179	(1)
Linear classification of C(K) spaces	180	(4)
Renormings of C(K) spaces	184	(5)
Nonlinear classification of C(K) spaces	189	(12)
References	194	(7)
Metrizable spaces and generalizations	201	(26)
G. Gruenhage
Introduction	203	(1)
Metrics, metrizable spaces, and mappings	203	(2)
Networks	205	(2)
Monotone normality	207	(1)
Stratifiable and related spaces	208	(2)
Some higher cardinal generalizations	210	(1)
Moore and developable spaces	211	(1)
Bases with certain order properties	212	(2)
Normality in products	214	(1)
Sums of metrizable subspaces	215	(1)
Open problems	216	(11)
References	219	(8)
Recent progress in the topological theory of semigroups and the Algebra of βS	227	(26)
N. Hindman
D. Strauss
Introduction	229	(2)
Topological and semitopological semigroups	231	(1)
Right (or left) topological semigroups	232	(4)
Algebra of βS	236	(3)
Applications to Ramsey Theory	239	(3)
Partial semigroups	242	(11)
References	244	(9)
Recent progress in hyperspace topologies	253	(34)
L'. Hola
J. Pelant
Preliminaries	255	(2)
Cardinal invariants of hyperspaces	257	(6)
Consonance	263	(5)
Generalized metric properties of hyperspaces	268	(4)
Completeness properties of hyperspaces	272	(5)
Compactness in hyperspaces	277	(10)
References	279	(8)
Some topics in geometric topology	287	(26)
K. Kawamura
Introduction	289	(1)
Generalized manifolds and the recognition problem of topological manifolds	289	(2)
Cohomological dimension theory	291	(2)
Compactifications in geometric topology	293	(7)
Approximate fibrators	300	(2)
Some other topics	302	(11)
References	304	(9)
Quasi-uniform spaces in the year 2001	313	(32)
H.-P. Kunzi
Introduction	315	(2)
Basic concepts	317	(3)
Extensions and completions	320	(6)
Functorial quasi-uniformities	326	(1)
Quasi-pseudometric spaces	327	(3)
Uniformizable ordered spaces	330	(1)
Hyperspaces and (multi)function spaces	331	(2)
Topological algebra	333	(12)
References	336	(9)
Functions spaces	345	(26)
W. Marciszewski
Introduction	347	(1)
Function spaces on metrizable spaces	348	(7)
Function spaces on countable spaces	355	(4)
Products of function spaces	359	(3)
Condensations of function spaces	362	(1)
Miscellaneous results	363	(8)
References	364	(7)
Topology and domain theory	371	(24)
K. Martin
M.W. Mislove
G.M. Reed
Introduction	373	(2)
Domain theory	375	(8)
Models of topological spaces	383	(4)
Measurement	387	(8)
References	391	(4)
Topics in dimension theory	395	(28)
R. Pol
H. Torunczyk
Introduction	397	(1)
Terminology	397	(1)
Weakly infinite-dimensional spaces and Haver's property C	398	(2)
Extension theory	400	(2)
Products of compact spaces	402	(1)
Products of non-compact spaces	403	(1)
Hereditarily indecomposable continua in dimension theory	404	(2)
Pushing compacta off affine manifolds in Euclidean spaces	406	(1)
Basic embeddings	407	(1)
Transfinite dimensions	407	(2)
The gap between the dimensions	409	(1)
Dimension-raising mappings with lifting properties	410	(1)
Universal spaces	411	(1)
Miscellaneous topics	412	(11)
References	415	(8)
Continuous selections of multivalued mappings	423	(40)
D. Repovs
P.V. Semenov
Solution of Michael's problem for C-domains	425	(6)
Selectors for hyperspaces	431	(6)
Relations between U- and L-theories	437	(7)
Miscellaneous results	444	(8)
Open problems	452	(11)
References	456	(7)
Convergence in the presence of algebraic structure	463	(22)
D. Shakhmatov
Definitions of main convergence properties	465	(1)
Convergence properties in topological spaces	466	(1)
Convergence properties in topological groups	467	(3)
Convergence properties in groups with additional compactness conditions	470	(2)
Convergence properties in functions spaces Cp(X)	472	(1)
Convergence properties in products	473	(3)
Sequential order in topological groups and function spaces	476	(1)
Miscellaneous	477	(8)
References	480	(5)
Descriptive set theory in topology	485	(30)
S. Solecki
Introduction	487	(1)
Polish topological group actions	488	(11)
Topologies on groups and ideals and complexity of their actions	499	(7)
Composants in indecomposable continua	506	(3)
Classifications of topological objects	509	(6)
References	511	(4)
Topological groups: between compactness and No-boundedness	515	(30)
M. Tkachenko
Introduction	517	(1)
Countably compact groups	518	(7)
O-bounded and strictly o-bounded groups	525	(3)
R-factorizable groups	528	(17)
References	540	(5)
Essays	545	(32)
Anderson, R.D., The early development of infinite dimensional topology	547	(2)
Comfort, W.W., Topological combinatorics: A peaceful pursuit	549	(4)
Henriksen, M., Topology related to rings of real-valued continuous functions. Where it has been and where it might be going	553	(4)
Mardesic, S., Shape theory	557	(4)
Nagata, J., Looking back at modern general topology in the last century	561	(4)
Rudin, M.E., Topology in the 20th Century	565	(4)
Smirnov, Yu.M., Compact extensions	569	(4)
Reminiscences of L. Vietoris	573	(4)
List of open problems and questions	577	(22)
Arhangel'skii, A.V.	579	(4)
Bennett, H.R. and D.J. Lutzer	583	(1)
Dijkstra, J. and J. van Mill	584	(1)
Godefroy, G.	584	(1)
Gruenhage, G.	585	(2)
Hindman, N. and D. Strauss	587	(1)
Kawamura, K.	587	(1)
Kunzi, H.-P.	588	(1)
Marciszewski, W.	588	(2)
Martin, K., M.W. Mislove and G.M. Reed	590	(1)
Pol, R. and H. Torunczyk	590	(1)
Repovs, D. and P.V. Semenov	591	(2)
Shakhmatov, D.	593	(2)
Solecki, S.	595	(1)
Tkachenko, M.	596	(3)
Author index	599	(12)
Special symbols	611	(2)
Subject index	613