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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Baire reflection


Authors: Stevo Todorcevic and Stuart Zoble
Journal: Trans. Amer. Math. Soc. 360 (2008), 6181-6195
MSC (2000): Primary 03E55; Secondary 03E50
Published electronically: July 24, 2008
MathSciNet review: 2434283
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Abstract: 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.


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Additional Information

Stevo Todorcevic
Affiliation: Department of Mathematics, University of Toronto, Toronto, Canada M5S 2E4 – and – Universite Paris 7-CNRS, UMR 7056, 2 Place Jussieu, 75251 Paris Cedex 05, France
Email: stevo@math.toronto.edu

Stuart Zoble
Affiliation: Department of Mathematics, University of Toronto, Toronto, Canada M5S 2E4
Address at time of publication: Department of Mathematics, Wesleyan University, 265 Church Street, Middletown, Connecticut 06459-0128
Email: azoble@wesleyan.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04503-0
PII: S 0002-9947(08)04503-0
Keywords: Baire Property, Game Reflection, Martin's Maximum
Received by editor(s): March 10, 2006
Published electronically: July 24, 2008
Article copyright: © Copyright 2008 American Mathematical Society