Handbook of Set-Theoretic Topology (eBook)

Front Cover 1
Handbook of Set-Theoretic Topology 4
Copyright 
5 
Foreword 6
Table of Contents 8
CHAPTER 1. Cardinal Functions I 10
Introduction 12
1. Notation and definitions 14
2. Combinatorial principles 16
3. Definitions of cardinal functions and elementary inequalities 19
4. Bounds on the cardinality of X 26
5. Bounds using spread 31
6. Bounds using cellularity and 
33 
7. Cardinal functions on compact spaces 34
8. Cardinal functions on metrizable spaces 41
9. Bounds on the number of compact sets in X bounds using extent
10. Bounds on the number of continuous, real-valued functions on X 48
11. Density and cellularity of product spaces 50
12. Achieving cellularity and spread 54
13. The cardinal number o(X) and related results 56
14. Examples 59
15. Summary of definitions and inequalities 62
References 66
Chapter 2. Cardinal functions 
72 
Introduction 74
1. The sharpness of bounds on the cardinality of X 74
2. The sharpness of bounds using spread 83
3. On products 89
4. Subspaces of compact spaces 91
5. Subspaces of Lindelöf spaces 98
6. Omitting cardinals 102
7. The character of w1 in first countable 
107 
8. On sup 
114 
References 117
Chapter 3. The integers and topology 120
1. Introduction 122
2. Convention and notation 123
3. Six cardinals 124
4. The cardinal 
133 
5. Consistency results 136
6. Sequential compactness and countable compactness 137
7. Spaces from C*-chains 142
8. Separable metrizable spaces 145
9. G& 's and
157 
10. Covering properties and the Michael line 159
11. Spaces from almost disjoint families 162
12. Weakenings of normality 164
13. Recursive constructions of topologies 168
References 173
Chapter 4. Box 
178 
Introduction 180
1. Preliminaries and elementary results 180
2. Paracompact spaces 186
3. The nabla lemma (for countable products 
190 
4. Large compact factors 193
5. CH and lighter axioms 197
6. In forcing extensions 202
References 207
Chapter 5. Special subsets of the real 
210 
1. Introduction 212
2. Luzin and Sierpinski sets 213
3. Concentrated sets and sets of strong measure zero 216
4. s-sets 
219 
5. Universal measure zero sets, perfectly meager sets, ., . and s0-sets, and Hausdorff 
221 
6. Order types of the real line 227
7. Unions 228
8. Products 229
9. Continuous and homeomorphic images and C" and C'-sets 231
10. Implications and definitions 236
References 237
Chapter 6. Trees and linearly ordered sets, 244
Introduction 246
1. Preliminaries 246
2. Trees 248
3. Linearly ordered sets 256
4. A class of 
262 
5. Aronszajn trees and lines 265
6. Suslin trees and lines 275
7. Aronszajn and Suslin trees of greater heights 282
8. Kurepa trees and lines 285
9. Nonspecial trees 293
References 298
Chapter 7. Basic S and 
304 
1. Introduction 306
2. Prerequisites 308
3. Canonical S- and L-spaces 310
4. CH constructions 314
5. Applications 318
6. Destroying S and L using MA 4+ 
323 
7. Two theorems about S-spaces 328
References 333
Chapter 
336 
Introduction 338
1. Consistency of first-countable S-spaces 339
2. Martin's Axiom and first-countable L-spaces 344
3. Some methods for getting models of Martin's Axiom 344
4. S-spaces with strong combinatorial properties 347
5. 2-complicated spaces 348
6. L-spaces and Martin's Axiom 353
References 354
Chapter 9. Covering 
356 
1. Introduction 358
2. Characterizations of paracompact spaces 360
3. Definitions and characterizations of other covering properties 369
4. Relationships among covering properties 379
5. Mappings and covering properties 393
6. Products 398
7. Subspaces and sums 411
8. With locally compact or locally connected 414
Chapter 9. With countably compact or pseudocompact 422
References 425
Chapter 10. Generalized metric 
432 
Introduction 434
1. Review of basic metrization theory 
435 
2. Gd-diagonals and submetrizable 
437 
3. Simultaneous generalizations of metric 
444 
4. Networks: 
454 
5. Stratifiable and related 
462 
6. Base of 
477 
7. Spaces with a point-countable 
481 
8. Quasi-developable 
486 
9. Semi-metrizable and symmetrizable spaces 489
10. Quasi-metrizable and related spaces 497
11. k-networks: 
502 
References 506
Index 510
Chapter 11. An introduction to 
512 
Introduction 514
0. Preliminaries 515
1. The spaces ßw 
517 
2. The spaces ßw and ßw/w 
536 
3. Partial orderings on 
548 
4. Weak P-points and other points in 
557 
5. Remarks 571
Open problems 572
References 573
Chapter 12. Countably compact and 
578 
1. Introduction 580
2. Basic properties and examples 580
3. When is a product of countably compact spaces countably compact? 585
4. Products and r-limits 593
5. When is a countably compact or compact space sequentially compact? 598
6. When is a countably compact space compact? 602
7. Notes 607
References 609
Chapter 13. Initially k-compact and related 
612 
1. Introduction 614
2. Initial k- and [., k]-compactness 
615 
3. Basic properties of initially k-compact 
617 
4. Examples of initially 
622 
5. Products of initially 
628 
6. Further examples, product theorems, and open problems 632
7. Notes 638
References 639
Chapter 14. The theory of nonmetrizable 
642 
1. Introduction 644
2. Cardinal invariants, covering and separation properties 646
3. Examples 651
4. Martin's Axiom and Type I spaces 664
5. The structure of 
671 
6. A variety of 
677 
References 691
Chapter 15. Normality versus collectionwise 
694 
Introduction 696
I. 
697 
II. 
716 
III. 
730 
IV. Historical 
732 
References 737
Chapter 
742 
1. Introduction, notations, definitions 744
2. Nyikos' theorem 746
3. Strongly compact cardinals, random reals, Con(PMEA) 748
4. Summary and an alternative proof 753
5. Normal, not collectionwise normal spaces 756
6. Navy's space 759
7. The CH example 762
8. Large cardinals 
766 
References 768
Chapter 17. Dowker 
770 
1. Dowker's theorem 772
2. Classes of spaces which cannot be Dowker 774
3. Some Dowker spaces 776
4. Related topics 782
References 787
Chapter 18. Products of normal spaces, 790
1. Introduction 792
2. Products of normal spaces need not be normal 793
3. Products with a compact factor 799
4. Products with a metric factor 809
5. Hereditarily normal products 820
6. Infinite products 822
7. 
828 
8. Some open problems 830
References 832
Chapter 19. Versions of 
836 
0. Introducing 
838 
1. The topology of 
839 
2. MA 
843 
3. Partial 
846 
4. The influence 
853 
5. MA for restricted kinds of 
856 
6. Extending a little extending a lot
7. 
867 
8. Axiomatic solidarity 873
9. Would you like some proof? 879
References 893
Chapter 20. Random and Cohen 
896 
0. Introduction 898
1. Properties of the meagre and null ideals 898
2. Cardinality questions 908
3. Forcing 909
4. Conclusion 920
References 920
Chapter 21. Applications 
922 
0. 
924 
1. Stationary sets, elementary substructures, and forcing 925
2. How to recognize a proper partial ordering 930
3. The Proper Forcing Axiom. Examples and counterexamples 933
4. Gaps and the cardinal-collapsing trick 939
5. TOP and the closed-unbounded-set trick 944
6. The real numbers and the Continuum-Hypothesis trick 950
7. Trees, . and 
957 
8. Reflection principles and PFA+ 962
9. Concluding remarks 965
References 967
Chapter 22. Borel measures, 970
1. Introduction 972
2. Conventions and definitions 973
3. Restrictions and extensions of Borel measures 975
4. Set-theoretic preliminaries 978
5. Some examples 982
6. Regularity and t-additivity 
991 
7. Completeness properties for finite Borel measures 994
8. Completeness properties for 2-valued Borel measures 1001
9. Completeness properties of small spaces 1004
10. Completeness and covering properties 1007
11. Radon spaces 1013
12. Regularity of s-finite Borel 
1023 
13. s-finiteness of diffused, outer 
1029 
14. Baire measures 1038
15. Products of real lines and Radon measures 1045
References 1049
Chapter 23. Banach spaces and 
1054 
Introduction 1056
0. Preliminaries 1057
1. Rosenthal's theorem for isomorphic embedding of 
1066 
2. Calibers, independent families of compact spaces K and universal isomorphic embeddings of l1a into 
1075 
3. Isomorphic embeddings of l1a in Banach spaces X and independent families on the dual 
1080 
4. Large subsets of L8(µ) far apart in L1-norm and Pelczynski's 
1084 
5. The Kunen-Haydon-Talagrand example 1095
6. Corson-compact spaces and subclasses—Applications to Banach spaces 1108
7. Kunen's example of an S-space and Banach spaces 1132
8. Fixed points of contractions in weakly compact convex subsets of Banach spaces 1138
References 1147
Chapter 24. Topological 
1152 
1. Introduction: Seven major results 1154
2. Topological pathology (upper bounds) 1167
3. Cardinal invariants 1172
4. Measurability conditions and the group Hom(G, T ) 1181
5. Topological groups with special topological properties 1199
6. Pseudocompact topological groups 1211
7. Dense subgroups with special properties 1222
8. Products of topological groups 1229
9. Topologizing a group 1235
10. Not all homogeneous spaces are topological groups 1252
11. Concluding remarks and acknowledgements 1256
References 1259
Index 1274