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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 boundedne...
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| Main Authors: | |
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| Corporate Authors: | |
| Group Author: | ; ; ; ; |
| Published: |
World Scientific Pub. Co.,
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| Publisher Address: | Singapore ; Hackensack, N.J. : |
| Publication Dates: | 2014. |
| Literature type: | eBook |
| Language: | English |
| Series: |
Lecture notes series, Institute for Mathematical Sciences, National University of Singapore ;
v. 26 |
| Subjects: | |
| Online Access: |
http://www.worldscientific.com/worldscibooks/10.1142/9013#t=toc |
| Summary: |
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 than on the relationship between different forcing axioms or their consistency strengths. |
| Carrier Form: | 1 online resource (xiii,219pages) . |
| Bibliography: | Includes bibliographical references (pages 217-219) |
| ISBN: | 9789814571586 |
| Index Number: | QA9 |
| CLC: | O141.4 |
| Contents: | 1. Baire category theorem and the Baire category numbers. 1.1. The Baire category method - a classical example. 1.2. Baire category numbers. 1.3. P-clubs. 1.4. Baire category numbers of posets. 1.5. Proper and semi-proper posets -- 2. Coding sets by the real numbers. 2.1. Almost-disjoint coding. 2.2. Coding families of unordered pairs of ordinals. 2.3. Coding sets of ordered pairs. 2.4. Strong coding. 2.5. Solovay's lemma and its corollaries -- 3. Consequences in descriptive set theory. 3.1. Borel isomorphisms between Polish spaces. 3.2. Analytic and co-analytic sets. 3.3. Analytic and co-analytic sets under p > 1 -- 4. Consequences in measure theory. 4.1. Measure spaces. 4.2. More on measure spaces -- 5. Variations on the Souslin hypothesis. 5.1. The countable chain condition. 5.2. The Souslin hypothesis. 5.3. A selective ultrafilter from m > 1. 5.4. The countable chain condition versus the separability -- 6. The S-spaces and the L-spaces. 6.1. Hereditarily separable and hereditarily Lindel f spaces. 6.2. Countable tightness and the S- and L-space problems -- 7. The side-condition method. 7.1. Elementary submodels as side conditions. 7.2. Open graph axiom -- 8. Ideal dichotomies. 8.1. Small ideal dichotomy. 8.2. Sparse set-mapping principle. 8.3. P-ideal dichotomy -- 9. Coherent and Lipschitz trees. 9.1. The Lipschitz condition. 9.2. Filters and trees. 9.3. Model rejecting a finite set of nodes. 9.4. Coloring axiomfor coherent trees -- 10. Applications to the S-space problem and the von Neumann problem. 10.1. The S-space problem and its relatives. 10.2. The P-ideal dichotomy and a problem of von Neumann -- 11. Biorthogonal systems. 11.1. The quotient problem. 11.2. A topological property of the dual ball. 11.3. A problem of Rolewicz. 11.4. Function spaces -- 12. Structure of compact spaces. 12.1. Covergence in topology. 12.2. Ultrapowers versus reduced powers. 12.3. Automatic continuity in Banach algebras -- 13. Ramsey theory on ordinals. 13.1. The arrow notation. 13.2. 2[symbol]. 13.3. 1[symbol] -- 14. Five cofinal types. 14.1. Tukey reductions and cofinal equivalence. 14.2. Directed posets of cardinality at most [symbol]. 14.3. Directed sets of cardinality continuum -- 15. Five linear orderings. 15.1. Basis problem for uncountable linear orderings. 15.2. Separable linear orderings. 15.3. Ordered coherent trees. 15.4. Aronszajn orderings -- 16. Cardinal arithmetic and mm. 16.1. mm and the continuum. 16.2. mm and cardinal arithmetic above the continuum -- 17. Reflection principles. 17.1. Strong reflection of stationary sets. 17.2. Weak reflection of stationary sets. 17.3. Open stationary set-mapping reflection. |