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Register a new userAlgebra, selections, and additive Ramsey theory
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Authors
Boaz Tsaban
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Hindmans celebrated Finite Sums Theorem, and its high-dimensional version due to Milliken and Taylor, are extended from covers of countable sets to covers of arbitrary topological spaces with Mengers classic covering property. The methods include, in addition to Hurewiczs game theoretic characterization of Mengers property, extensions of the classic idempotent theory in the Stone--Czech compactification of semigroups, and of the more recent theory of selection principles. This provides stro
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Assume that X is a metrizable separable space, and each clopen-valued lower semicontinuous multivalued map Phi from X to Q has a continuous selection. Our main result is that in this case, X is a sigma-space. We also derive a partial converse implication, and present a reformulation of the Scheepers Conjecture in the language of continuous selections.
A surprising number of new results in core SPM in the last quarter of 2007, and some other beautiful fundamental results are announced.
1. A Wikipedia entry on topological games 2. On a fragment of the universal Baire property for sigma^1_2 sets 3. The coarse classification of homogeneous ultra-metric spaces 4. Ramsey-like embeddings 5. Proper and piecewise proper families of reals 6. Measures and their random reals 7. Obtainable Sizes of Topologies on Finite Sets 8. Spaces of R-places of rational function fields 9. All properties in the Scheepers Diagram are linearly-sigma-additive.
Contents: 1. Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces, I; 2. Frechet-Urysohn fans in free topological groups; 3. Packing index of subsets in Polish groups; 4. Symmetric monochromatic subsets in colorings of the Lobachevsky plane; 5. Structural Ramsey theory of metric spaces and topological dynamics of isometry groups; 6. Distinguishing Number of Countable Homogeneous Relational Structures; 7. Indestructible colourings and rainbow Ramsey theorems; 8. Products of Borel subgroups; 9. Selection theorems and treeability; 10. Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces, IV; 11. A property of Cp[0, 1]; 12. A Dedekind Finite Borel Set; 13. Aronszajn Compacta; 14. A strong antidiamond principle compatible with CH; 15. On the strength of Hausdorffs gap condition; 16. Nonhomogeneous analytic families of trees; 17. Reasonable non-Radon-Nikodym ideals; 18. Continuity and related forcing; 19. An exact Ramsey principle for block sequences; 20. Baire reflection; 21. Tukey classes of ultrafilters on; 22. Countably determined compact abelian groups; 23. A topological reflection principle equivalent to Shelahs Strong Hypothesis; 24. Superfilters, Ramsey theory, and van der Waerdens Theorem.
Contents: 2. Invited contribution: Ultrafilters and small sets 3. Research announcements 3.1. Inverse Systems and I-Favorable Spaces 3.2. Combinatorial and hybrid principles for sigma-directed families of countable sets modulo finite 3.3. A dichotomy characterizing analytic digraphs of uncountable Borel chromatic number in any dimension 3.4. A dichotomy characterizing analytic digraphs of uncountable Borel chromatic number in any dimension 3.5. Large continuum, oracles 3.6. Borel hierarchies in infinite products of Polish spaces 3.7. A game for the Borel functions 3.8. On some problems in general topology 4. Problem of the Issue
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