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Record Nr. |
UNINA9910790973803321 |
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Autore |
Todorcevic Stevo |
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Titolo |
Notes on forcing axioms / / Stevo Todorcevic, University of Toronto, Canada ; editors, Chitat Chong, Qi Feng, Yue Yang, National University of Singapore, Singapore, Theodore A. Slaman, W. Hugh Woodin, University of California, Berkeley, USA |
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Pubbl/distr/stampa |
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New Jersey : , : World Scientific, , [2014] |
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�2014 |
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ISBN |
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Descrizione fisica |
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1 online resource (xiii, 219 pages) : illustrations |
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Collana |
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Lecture notes series (Institute for Mathematical Sciences, National University of Singapore), , 1793-0758 ; ; volume 26 |
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Disciplina |
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Soggetti |
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Forcing (Model theory) |
Axioms |
Baire classes |
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Lingua di pubblicazione |
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Formato |
Materiale a stampa |
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Livello bibliografico |
Monografia |
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Note generali |
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Description based upon print version of record. |
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Nota di bibliografia |
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Includes bibliographical references. |
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Nota di contenuto |
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Contents; Foreword by Series Editors; Foreword by Volume Editors; Preface; 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 > ω1; 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 > ω1; 5.4 The countable chain condition versus the separability; 6 The S-spaces and the L-spaces; 6.1 Hereditarily separable and hereditarily Lindelof spaces; 6.2 Countable tightness and the S- and L-space problems; 7 The Side-condition Method; 7.1 Elementary submodels as side conditions |
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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 axiom for 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 |
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; Appendix A Basic Notions; A.1 Set theoretic notions; A.2 Δ-systems and free sets; A.3 Topological notions; A.4 Boolean algebras; Appendix B Preserving Stationary Sets; B.1 Stationary sets; B.2 Partial orders, Boolean algebras and topological spaces; B.3 A topological translation of stationary set preserving |
Appendix C Historical and Other Comments |
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Sommario/riassunto |
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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 |
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