Open Problems in Topology II (eBook)

Elliott M. Pearl (Herausgeber)

eBook Download: PDF

2011 | 1. Auflage
776 Seiten
Elsevier Science (Verlag)
978-0-08-047529-5 (ISBN)

Infos

This volume is a collection of surveys of research problems in topology and its applications. The topics covered include general topology, set-theoretic topology, continuum theory, topological algebra, dynamical systems, computational topology and functional analysis.

* New surveys of research problems in topology
* New perspectives on classic problems
* Representative surveys of research groups from all around the world

Inhalt

Cover 1
Copyright Page 5
Preface 6
Table of Contents 8
Part 1. General Topology 14
Chapter 1. Selected ordered space problems 16
1. Introduction 16
2. A few of our favorite things 16
References 19
Chapter 2. Problems on star-covering properties 22
Introduction 22
Compactness-type properties 22
Lindelöf-type properties and cardinal functions 23
Paracompactness-type properties 23
Property (a) 24
References 25
Chapter 3. Function space topologies 28
1. Splitting and admissible topologies 28
2. The greatest splitting, compact-open, and Isbell topologies 30
References 33
Chapter 4. Spaces and mappings: special networks 36
Introduction 36
Mappings 36
k-networks and weak bases 39
Spaces determined or dominated by certain covers 40
Products of k-spaces having certain k-networks 42
References 45
Chapter 5. Extension problems of real-valued continuous functions 48
1. Introduction 48
2. C-embedding versus C*-embedding 48
3. p-embedding and pZ-embeddings 52
4. Miscellaneous questions 55
References 56
Chapter 6. LS(k)-spaces 60
References 63
Chapter 7. Problems on (ir)resolvability 64
Introduction. Resolvability hierarchy 64
Resolvability of connected spaces 64
Baire property and a problem of Katetov 65
Groups and homogeneous spaces 66
Compactness, products, miscellaneous questions 67
Structure resolvability 68
Literature 69
References 70
Chapter 8. Topological games and Ramsey theory 74
Introduction 74
1. Basic concepts 75
2. The point-picking ganes 76
3. Modifications of the point-picking games 84
4. The set-picking games 87
5. Further restrictions on ONE 89
6. Banach–Mazur games 90
7. The length of games 93
8. Memory restrictions on the winning player 94
9. Ramsey theory 97
References 99
Chapter 9. Selection principles and special sets of reals 104
1. Introduction 104
2. The Scheepers Diagram problem 105
3. Examples without special set-theoretic hypotheses 106
4. Examples from CH or MA 108
5. The d-property 109
6. Preservation of properties 109
7. Modern types of covers 111
8. Splittability 113
9. Function spaces and local-global principles 114
10. Topological groups 115
11. Cardinal characteristics of the continuum 117
12. Additional problems and other special sets of reals 118
References 119
Part 2. Set-theoretic Topology 122
Chapter 10. Introduction: Twenty problems in set-theoretic topology 124
References 126
Chapter 11. Thin-tall spaces and cardinal sequences 128
1. Introduction 128
2. Spaces of countable height 129
3. Spaces of height < .2
4. Spaces of height .2 131
5. Spaces of height > .2
6. A basic construction 133
7. Forcing an LCS space 133
8. Final remarks 135
References 136
Chapter 12. Sequential order 138
References 140
Chapter 13. On D-spaces 142
Introduction 142
Questions about D-spaces 142
Stickyness 144
General remarks 146
References 146
Chapter 14. The fourth head of ßN 148
1. Trivial continuous maps 148
2. A partial result 150
3. Rigidity phenomena for quotients P(N)/I 150
4. Further results 151
5. Conclusion 152
References 154
Chapter 15. Are stratifiable spaces M1? 156
1. Equivalent questions 157
2. Related classes and partial results 158
3. Function spaces and a possible counterexample 161
4. Some final remarks 161
References 162
Chapter 16. Perfect compacta and basis problems in topology 164
1. Perfect compacta 165
2. Uncountable spaces 167
3. Approaches, axiomatics, further reading 168
References 170
Chapter 17. Selection problems for hyperspaces 174
1. Weak selections and orderability 175
2. Topological well-ordering and selections 178
3. Selections and disconnectedness-like properties 179
References 182
Chapter 18. Efimov’s problem 184
Introduction 184
1. Attacking the problem 184
2. Counterexamples 186
3. Is there still a problem? 187
4. Larger cardinals 188
References 190
Chapter 19. Completely separable MAD families 192
1. The main problem 192
2. Topological connection 194
3. MAD families in forcing extensions 195
References 196
Chapter 20. Good, splendid and Jakovlev 198
1. The size of good and splendid spaces 198
2. Connections to other problems 199
References 201
Chapter 21. Homogeneous compacta 202
0. Introduction 202
1. Rudin’s problem 202
2. Van Douwen’s problem 203
3. Arhangelskii's problem 204
4. Continuous images of homogeneous compacta 205
5. Remarks 205
References 206
Chapter 22. Compact spaces with hereditarily normal squares 210
References 213
Chapter 23. The metrization problem for Fréchet groups 214
1. Introduction 214
2. The role of gaps 215
3. Other convergence properties 216
4. In search of a test model 218
References 218
Chapter 24. Cech–Stone remainders of discrete spaces 220
1. Introduction 220
2. Some basics 221
3. Some consequences of Axiom O 223
4. Implications for .1* 225
5. Some more open problems about Axiom O 226
6. Notes on Problem 2 227
References 228
Chapter 25. First countable, countably compact, noncompact spaces 230
1. Consistent good examples for Problem 1 231
2. Other consistent constructions for Problem 1 233
3. Arbitrarily large first countable, locally compact, countably compact spaces 234
4. Towards negative answers 236
References 237
Chapter 26. Linearly Lindelöf problems 238
Introduction 238
Local compactness 239
Sequentially linearly Lindelöf spaces 240
Discretely Lindelöf spaces 241
Estimating cardinality 241
The Hušek number 242
A consistent realcompact example 242
A consistent first countable example 243
Lindelöf problems 243
Acknowledgments 244
References 244
Chapter 27. Small Dowker spaces 246
References 251
Chapter 28. Reflection of topological properties to Aleph1 254
References 259
Chapter 29. The Scarborough–Stone problem 262
1. Introduction 262
2. How to construct a product that is not countably compact 263
3. The Ostaszewski construction 264
4. Franklin–Rajagopalan spaces 266
5. Martin’s axiom and proper forcing 266
6. Concluding comments 267
References 268
Part 3. Continuum Theory 270
Chapter 30. Questions in and out of context 272
References 275
Chapter 31. An update on the elusive fixed-point property 276
References 286
Chapter 32. Hyperspaces of continua 292
Introduction 292
Whitney and diameter maps 292
Cones, products and hyperspaces 293
Means 294
Fixed point property 295
Mappings between hyperspaces 296
Unicoherence of F2(S1) 297
Locating cells in hyperspaces 297
Unique hyperspaces 298
1. Miscellaneous problems 299
References 300
Chapter 33 Inverse limits and dynamical systems 302
1. Introduction 302
2. Characterization of chainability 302
3. Plane embedding 303
4. Inverse limits on [0, 1] 307
5. The property of Kelley 309
6. Inverse limits with upper semi-continuous bonding functions 309
7. Applications of inverse limits in Economics 310
References 313
Chapter 34. Indecomposable continua 316
Hereditary equivalence 316
Homogeneity 316
Epsilon-premaps 319
Fixed points 319
Maps of products 320
Homeomorphism groups 321
Q-like continua 322
Hyperspaces 323
Dimensions greater than one 324
Non-metric continua 325
Conclusion 327
References 327
Chapter 35. Open problems on dendroids 332
1. Introduction 332
2. The problem 332
3. Mappings on dendrites 332
4. Maps onto dendroids 337
5. Contractibility 337
6. Hyperspaces 339
7. Property of Kelley 339
8. Retractions 340
9. Means 341
10. Selections 342
11. Smooth dendroids 343
12. Planability 344
13. Shore sets 345
References 345
Chapter 36. 1/2-Homogeneous continua 348
1. Introduction 348
2. Notation and terminology 349
3. 1/2-Homogeneous continua with cut points 349
4. 1/2-Homogeneous cones 352
5. 1/2-Homogeneous hyperspaces 355
References 357
Chapter 37. Thirty open problems in the theory of homogeneous continua 358
1. Preliminaries 359
2. Fourteen miscellaneous problems 360
3. Filament sets: definitions and basic properties 362
4. Filament sets: sixteen questions 365
References 368
Part 4. Topological Algebra 370
Chapter 38. Problems about the uniform structures of topological groups 372
1. Introduction 372
2. The two versions of Itzkowitz’s problem 373
3. Some remarks about [FSIN] and [SIN] 374
4. The class [ASIN] 375
5. A few other questions 376
6. A representative case 377
References 378
Chapter 39. On some special classes of continuous maps 380
1. Special morphisms of Top 380
2. Corresponding morphisms in related categories 386
References 388
Chapter 40. Dense subgroups of compact groups 390
1. Introduction 390
2. Groups with topologies of pre-assigned type 391
3. Topologies induced by groups of characters 392
4. Extremal phenomena 395
5. Miscellaneous questions 396
References 399
Chapter 41. Selected topics from the structure theory of topological groups 402
1. Dimension theory of topological groups 402
2. Pseudocompact and countably compact group topologies on Abelian groups 403
3. Properties determined by convergent sequences 406
4. Categorically compact groups 408
5. The Markov–Zariski topology 409
6. Bohr topologies of Abelian groups 412
7. Miscellanea 415
References 416
Chapter 42. Recent results and open questions relating Chu duality and Bohr compactifications of locally compact groups 420
1. Introduction 420
2. Basic definitions 421
3. Abelian groups 423
4. Nonabelian groups 424
5. How is G placed in bG? Interpolation sets 427
Acknowledgement 433
References 433
Chapter 43. Topological transformation groups: selected topics 436
1. Introduction 436
2. Equivariant compactifications 436
3. Equivariant normality 439
4. Universal actions 440
5. Free topological G-groups 442
6. Banach representations of groups 442
7. Dynamical versions of Eberlein and Radon–Nikodým compacta 446
Acknowledgment 447
References 447
Chapter 44. Forty-plus annotated questions about large topological groups 452
References 461
Part 5. Dynamical Systems 464
Chapter 45. Minimal flows 466
1. Introduction 466
2. Minimal flows on 3-manifolds 468
3. Asymptotic properties 472
References 474
Chapter 46. The dynamics of tiling spaces 476
Topological rigidity 476
The topological structure of tiling spaces 477
Deformations of tiling spaces 478
Mixing properties 479
Tiling spaces that are not locally finite 479
Pisot conjecture 480
New directions 480
Acknowledgments 480
References 480
Chapter 47. Open problems in complex dynamics and complexŽ topology 482
1. Cantor Bouquets and Indecomposable Continua 482
2. Sierpinski Curve Julia Sets 487
References 490
Chapter 48. The topology and dynamics of flows 492
Flows 492
Templates for basic sets 494
Twist-wise flow equivalence 495
Putting the pieces together and realization problems 497
Bonatti’s geometric type 499
References 501
Part 6. Computer Science 504
Chapter 49. Computational topology 506
1. Introduction 506
2. History 507
3. Computation and the reals 508
4. Correctly embedded approximations for graphics & applications
5. The role for differentiability 519
6. Computational topology resolution 535
7. Computational topology and surface reconstruction 538
8. Computational topology and low-dimensional manifolds 539
9. Skeletal structures 542
10. Computational topology and Biology on simplicial complexes 543
11. Finite approximation and (non-Hausdorff) topology 545
12. Algorithmic topology and computational topology 546
13. Computational topology workshop of 1999 547
14. Conclusion 550
15. Acknowledgements 550
References 550
Part 7. Functional Analysis 560
Chapter 50. Non-smooth analysis, optimisation theory and Banach space theory 562
1. Weak Asplund spaces 562
2. The Bishop–Phelps problem 563
3. The complex Bishop–Phelps property 564
4. Biorthogonal sequences and support points 564
5. Best approximation 566
6. Metrizability of compact convex sets 567
7. The boundary problem 567
8. Separate and joint continuity 568
Acknowledgements 569
References 569
Chapter 51. Topological structures of ordinary differential equations 574
References 578
Chapter 52. The interplay between compact spaces and the Banach spaces of their continuous functions 580
Introduction 580
Subspaces 584
Quotients 585
Complemented subspaces 587
References 591
Chapter 53. Tightness and t-equivalence 594
References 596
Chapter 54. Topological problems in nonlinear and functional analysis 598
References 604
Chapter 55. Twenty questions on metacompactness in function spaces 608
1. Introduction 608
2. Notation and terminology 608
3. Metacompactness in Cp(X) for general spaces X 608
4. Metacompactness in Cp(X) when X is compact 610
References 611
Part 8. Dimension Theory 612
Chapter 56. Open problems in infinite-dimensional topology 614
Introduction 614
1. Higher-dimensional descriptive set theory 614
2. Zn-sets and related questions 616
3. The topology of convex sets and topological groups 618
4. Topological characterization of particular infinite-dimensional spaces 621
5. Problems on ANRs 622
6. Infinite-dimensional problems from Banach space theory 624
7. Some problems in dimension theory 629
8. Homological methods in dimension theory 629
9. Infinite-dimensional spaces in nature 632
References 634
Chapter 57. Classical dimension theory 638
Introduction 638
Coincidence of dimensions 639
Addition theorems for dimensions dim, Ind, ind 641
Product theorems for dimensions dim, ind, Ind 643
Compactifications 644
Infinite-dimensional theory 644
Compactness degrees 647
References 647
Chapter 58. Questions on weakly infinite-dimensional spaces 650
Introduction 650
1. Definitions 650
2. Maps, products, and subsets 653
3. Transfinite dimensions 654
References 657
Chapter 59. Some problems in the dimension theory of compacta 660
1. On the coincidence of dim, ind, Ind, and . for compact spaces 660
2. Noncoincidence of dim and ind for compact spaces 662
3. On the noncoincidence of ind and Ind for compact spaces 663
4. Dimensional properties of topological products 664
5. A problem concerning the subset theorem 664
References 665
Part 9. Invited Problems 666
Chapter 60. Problems from the Lviv topological seminar 668
Introduction 668
1. Asymptotic dimension 668
2. Extension of metrics 670
3. Questions in general topology 671
4. Some problems in Ramsey theory 672
5. Questions on functors in the category of compact Hausdorff spaces 675
References 678
Chapter 61. Problems from the Bizerte–Sfax–Tunis Seminar 682
Introduction 682
Spectral spaces and related topics 682
The space of leaves and the space of leaves classes 684
Dynamics of groups of homeomorphisms 685
Vector fields on surfaces 685
References 686
Chapter 62. Cantor set problems 688
Introduction 688
The problems 688
References 690
Chapter 63. Problems from the Galway Topology Colloquium 692
1. Universals: an introduction 692
2. Embedding ordering among topological spaces: an introduction 696
3. Questions relating to countable paracompactness 698
4. Abstract dynamical systems 700
References 701
Chapter 64. The lattice of quasi-uniformities 704
Introduction 704
Adjacent quasi-uniformities 705
Complements 708
References 709
Chapter 65. Topology in North Bay: some problems in continuum theory, dimension theory and selections 710
1. Introduction 710
2. Problems in dimension theory 710
3. Selections and C-property 714
4. Parametrization of the disjoint n-disks property 716
5. Locally connected continua 717
References 721
Chapter 66. Moscow questions on topological algebra 724
1. Topological groups 724
2. Maltsev spaces and retracts of groups 730
3. Convex compact spaces and affine functions 734
4. Stratifiable function spaces 735
5. Semilattices of compact G-extensions 735
References 736
Chapter 67. Some problems from George Mason University 740
Introduction 740
Kulesza’s problems 740
Levy’s problems 741
Matveev’s problems 742
References 743
Chapter 68. Some problems on generalized metrizable spaces 744
Sequence-covering maps 744
s-spaces and S-spaces 746
Aleph0-spaces 747
References 748
Chapter 69. Problems from the Madrid Department of Geometry and Topology 750
References 754
Chapter 70. Cardinal sequences and universal spaces 756
References 759
List of contributors 760
Index 764

Produkt-Details

• Erscheint lt. Verlag: 11.8.2011
• Sprache: englisch
• Themenwelt: Sachbuch/Ratgeber
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 0-08-047529-9 / 0080475299
• ISBN-13: 978-0-08-047529-5 / 9780080475295