Baire reflection

We study reflection principles involving nonmeager sets and the Baire Property which are consequences of the generic supercompactness of \omega _2, such as the principle asserting that any point countable Baire space has a stationary set of closed subspaces of weight \omega _1 which are also Baire s...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2008-12, Vol.360 (12), p.6181-6195
Main Authors: Todorcevic, Stevo, Zoble, Stuart
Format: Article
Language:English
Subjects:
Cardinal points
Exact sciences and technology
Games
General topology
Homeomorphism
Increasing sequences
Mathematical logic
Mathematical sets
Mathematical theorems
Mathematics
Partially ordered sets
Sciences and techniques of general use
Stationary sets
Topology
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
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Summary:We study reflection principles involving nonmeager sets and the Baire Property which are consequences of the generic supercompactness of \omega _2, such as the principle asserting that any point countable Baire space has a stationary set of closed subspaces of weight \omega _1 which are also Baire spaces. These principles entail the analogous principles of stationary reflection but are incompatible with forcing axioms. Assuming MM, there is a Baire metric space in which a club of closed subspaces of weight \omega _1 are meager in themselves. Unlike stronger forms of Game Reflection, these reflection principles do not decide CH, though they do give \omega _2 as an upper bound for the size of the continuum.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-08-04503-0