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The Hurewicz dichotomy for generalized Baire spaces

Monday 24th August 2015 - 16:00 to 17:00
INI Seminar Room 1
By classical results of Hurewicz, Kechris and Saint-Raymond, an analytic subset of a Polish space X is covered by a Ksigma subset of X if and only if it does not contain a closed-in-X subset homeomorphic to the Baire space omega^omega. We consider the analogous statement (which we call Hurewicz dichotomy) for Sigma11 subsets of the generalized Baire space kappa^kappa for a given uncountable cardinal kappa with kappa=kappa^(<kappa), and show how to force it to be true in a cardinal and cofinality preserving extension of the ground model. Moreover, we show that if the GCH holds, then there is a cardinal preserving class forcing extension in which the Hurewicz dichotomy for Sigma11 subsets of kappa^kappa holds at all uncountable regular cardinals kappa, while strongly unfoldable and supercompact cardinals are preserved. On the other hand, in the constructible universe L the dichotomy for Sigma11 sets fails at all uncountable regular cardinals, and the same happens in any generic extension obtained by adding a Cohen real to a model of GCH. This is joint work with Philipp Lücke and Luca Motto Ros.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons