The Hurewicz dichotomy for generalized Baire spaces
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| Description: |
Schlicht, P (Universität Bonn)
Monday 24 August 2015, 16:00-17:00 |
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| Created: | 2015-08-25 17:50 |
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| Collection: | Mathematical, Foundational and Computational Aspects of the Higher Infinite |
| Publisher: | Isaac Newton Institute |
| Copyright: | Schlicht, P |
| Language: | eng (English) |
| Distribution: |
World
(downloadable)
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| Explicit content: | No |
| Aspect Ratio: | 16:9 |
| Screencast: | No |
| Bumper: | UCS Default |
| Trailer: | UCS Default |
| Abstract: | 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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