Hausdorff Gaps and Limits (eBook)

Front Cover 1
Hausdorff Gaps and Limits 4
Copyright Page 5
Table of Contents 9
Notation and terminology 12
Chapter 1. Boolean Algebras 16
1 Introduction 16
2 Formulas 17
3 Atoms 20
4 Complete algebras 21
5 Homomorphism and filters 24
6 Ultrafilters 30
7 Extending a homomorphism 32
8 Chains and antichains 35
Problems 39
Chapter 2. Gaps and Limits 46
1 Introduction 46
2 Dominance 48
3 Hausdorff Gaps 53
4 The Parovicenko theorem 56
5 Types of gaps and limits 61
Problems 71
Chapter 3. Stone Spaces 78
1 The Stone representation 78
2 Subalgebras and homomorphisms 81
3 Zero-sets 82
4 The Stone-Cech compactification 85
5 Spaces of uniform ultrafilters 89
6 Strongly zero-dimensional spaces 93
7 Extremally disconnected spaces 96
Problems 100
Chapter 4. F-Spaces 108
1 Extending a function 108
2 Characterization of countable gaps 109
3 Construction of Parovicenko spaces 111
4 Closed sets in the space 117
5 On the Parovicenko theorem 118
6 On P-sets in the space 121
7 Character of points 129
Problems 134
Chapter 5. p-Base Matrix 138
1 Base tree 138
2 Stationary sets 143
3 c-Points 146
Problems 156
Chapter 6. Inhomogeneity 158
1 Kunen's points 158
2 A matrix of independent sets 159
3 Countable sets in F-spaces 164
4 Inhomogeneity of products of compact spaces 168
Problems 174
Chapter 7. Extending of Continuous Functions 176
1 Weak Lindelöf property 176
2 A long convergent sequence 178
3 Strongly discrete sets 181
Problems 190
Chapter 8. The Martin Axiom 194
1 Continuous images 194
2 The space /ß[w1] 195
3 On the Parovicenko theorem 200
4 Gaps 206
5 Homomorphisms of C ( X ) 211
Problems 220
Chapter 9. Partitions of Antichains 224
1 Partition algebras 224
2 Complete algebras 227
3 Partition algebras under MA 230
4 More on partition algebras 242
Problems 253
Chapter 10. Small P-Sets in . 258
1 Proper forcing 258
2 On P-filters with the ccc 261
Problems 270
Chapter 11. Forcing 274
1 Set theory and its models 274
2 Forcing 280
3 Complete ernbeddings 284
4 Cardinal numbers 286
5 Selected models 288
6 Iterated forcing 294
7 The Martin Axiom 304
Bibliography 309
Index 312