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E-grāmata: Notes On Forcing Axioms

Edited by (Univ Of California, Berkeley, Usa), Edited by (Nus, S'pore), Edited by (Nus, S'pore), Edited by (Chinese Academy Of Sciences, China), (Univ Of Toronto, Canada), Edited by (The Univ Of California, Berkeley, Usa)
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Notes On Forcing AxiomsPalielināt
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In the mathematical practice, the Baire category method is a tool for establishing the existence of a rich array of generic structures. However, in mathematics, the Baire category method is also behind a number of fundamental results such as the Open Mapping Theorem or the Banach–Steinhaus Boundedness Principle. This volume brings the Baire category method to another level of sophistication via the internal version of the set-theoretic forcing technique. It is the first systematic account of applications of the higher forcing axioms with the stress on the technique of building forcing notions rather that on the relationship between different forcing axioms or their consistency strengths.
Foreword by Series Editors ix
Foreword by Volume Editors xi
Preface xiii
1 Baire Category Theorem and the Baire Category Numbers
1(12)
1.1 The Baire category method -- a classical example
1(2)
1.2 Baire category numbers
3(1)
1.3 P-clubs
4(2)
1.4 Baire category numbers of posets
6(2)
1.5 Proper and semi-proper posets
8(5)
2 Coding Sets by the Real Numbers
13(28)
2.1 Almost-disjoint coding
13(2)
2.2 Coding families of unordered pairs of ordinals
15(4)
2.3 Coding sets of ordered pairs
19(4)
2.4 Strong coding
23(8)
2.5 Solovay's lemma and its corollaries
31(10)
3 Consequences in Descriptive Set Theory
41(4)
3.1 Borel isomorphisms between Polish spaces
41(1)
3.2 Analytic and co-analytic sets
42(1)
3.3 Analytic and co-analytic sets under p > ω1
43(2)
4 Consequences in Measure Theory
45(6)
4.1 Measure spaces
45(3)
4.2 More on measure spaces
48(3)
5 Variations on the Souslin Hypothesis
51(10)
5.1 The countable chain condition
51(2)
5.2 The Souslin Hypothesis
53(1)
5.3 A selective ultrafilter from m > ω1
54(2)
5.4 The countable chain condition versus the separability
56(5)
6 The S-spaces and the L-spaces
61(12)
6.1 Hereditarily separable and hereditarily Lindelof spaces
61(3)
6.2 Countable tightness and the S- and L-space problems
64(9)
7 The Side-condition Method
73(8)
7.1 Elementary submodels as side conditions
73(2)
7.2 Open graph axiom
75(6)
8 Ideal Dichotomies
81(10)
8.1 Small ideal dichotomy
81(4)
8.2 Sparse set-mapping principle
85(3)
8.3 P-ideal dichotomy
88(3)
9 Coherent and Lipschitz Trees
91(12)
9.1 The Lipschitz condition
91(3)
9.2 Filters and trees
94(2)
9.3 Model rejecting a finite set of nodes
96(2)
9.4 Coloring axiom for coherent trees
98(5)
10 Applications to the S-space Problem and the von Neumann Problem
103(10)
10.1 The S-space problem and its relatives
103(3)
10.2 The P-ideal dichotomy and a problem of von Neumann
106(7)
11 Biorthogonal Systems
113(20)
11.1 The quotient problem
113(8)
11.2 A topological property of the dual ball
121(5)
11.3 A problem of Rolewicz
126(1)
11.4 Function spaces
127(6)
12 Structure of Compact Spaces
133(14)
12.1 Covergence in topology
133(4)
12.2 Ultrapowers versus reduced powers
137(6)
12.3 Automatic continuity in Banach algebras
143(4)
13 Ramsey Theory on Ordinals
147(22)
13.1 The arrow notation
147(1)
13.2 ω2 → (ω2, ω + 2)2
148(11)
13.3 ω1 → (ω1, α)2
159(10)
14 Five Cofinal Types
169(8)
14.1 Tukey reductions and cofinal equivalence
169(1)
14.2 Directed posets of cardinality at most N1
170(4)
14.3 Directed sets of cardinality continuum
174(3)
15 Five Linear Orderings
177(12)
15.1 Basis problem for uncountable linear orderings
177(1)
15.2 Separable linear orderings
177(4)
15.3 Ordered coherent trees
181(5)
15.4 Aronszajn orderings
186(3)
16 Cardinal Arithmetic and mm
189(4)
16.1 mm and the continuum
189(3)
16.2 mm and cardinal arithmetic above the continuum
192(1)
17 Reflection Principles
193(6)
17.1 Strong reflection of stationary sets
193(2)
17.2 Weak reflection of stationary sets
195(2)
17.3 Open stationary set-mapping reflection
197(2)
Appendix A Basic Notions
199(6)
A.1 Set theoretic notions
199(1)
A.2 Δ-systems and free sets
200(1)
A.3 Topological notions
201(1)
A.4 Boolean algebras
202(3)
Appendix B Preserving Stationary Sets
205(10)
B.1 Stationary sets
205(1)
B.2 Partial orders, Boolean algebras and topological spaces
206(4)
B.3 A topological translation of stationary set preserving
210(5)
Appendix C Historical and Other Comments
215(2)
Bibliography 217
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