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Baire reflection


Authors: Stevo Todorcevic and Stuart Zoble
Journal: Trans. Amer. Math. Soc. 360 (2008), 6181-6195
MSC (2000): Primary 03E55; Secondary 03E50
DOI: https://doi.org/10.1090/S0002-9947-08-04503-0
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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  • [1] Cummings, J., Large Cardinal Properties of Small Cardinals, 1988, unpublished manuscript
  • [2] Engelking, R., General Topology, Heldermann Verlag, 1989 MR 1039321 (91c:54001)
  • [3] Feng, Q., Jech, T., Local Clubs, Reflection, and Preserving Stationary Sets, Proceedings of the London Mathematical Society, Vol. 58, 1989, 237-257 MR 977476 (90a:03072)
  • [4] Feng, Q., Magidor, M., Woodin, H., Universally Baire Sets of Reals, MSRI Publications 26, 1992 MR 1233821 (94g:03095)
  • [5] Foreman, M., Todorcevic, S., A New Löwenheim-Skolem Theorem, Transactions of the American Mathematical Society 357, 2005, 1693-1715 MR 2115072 (2005m:03064)
  • [6] Fremlin, D.H., Consequences of Martin's Axiom, Cambridge University Press, 1984 MR 780933 (86i:03001)
  • [7] Galvin, F., Jech, T., Magidor, M., An Ideal Game, Journal of Symbolic Logic, Vol. 43, No. 2, 1978, 284-292 MR 0485391 (58:5237)
  • [8] Jech, T., Magidor, M., Mitchell, M., Prikry, K., Precipitous Ideals, Journal of Symbolic Logic, Vol. 45, No. 1, 1980, 1-8 MR 560220 (81h:03097)
  • [9] König, B., Generic Compactness Reformulated, Archive for Mathematical Logic 43, 2004, no. 3, 311-326 MR 2052885 (2005f:03079)
  • [10] Larson, P., Separating Stationary Reflection Principles, Journal of Symbolic Logic, Vol. 65, No. 1, 2000, 247-258 MR 1782117 (2001k:03094)
  • [11] Mitchell, W., Aronsajn Trees and the Independence of the Transfer Property, Annals of Mathematical Logic, Vol. 5, No. 1, 1975, 21-46 MR 0313057 (47:1612)
  • [12] Shelah, S., Cardinal Arithmetic, Vol. 29 of Oxford Logic Guides, Oxford University Press, 1994 MR 1318912 (96e:03001)
  • [13] Steel, J., $ PFA$ implies $ AD^{L(\mathbb{R})}$, Journal of Symbolic Logic, Vol. 45, No. 4, 2005, 1255-1296 MR 2194247
  • [14] Taylor, A.D., Regularity Properties of Ideals and Ultrafilters, Annals of Mathematical Logic 16, 1979, No. 1, 33-55 MR 530430 (83b:04003)
  • [15] Todorcevic, S., Localized Reflection and Fragments of PFA, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 58, 2002, 135-148 MR 1903856 (2003g:03081)
  • [16] Todorcevic, S., Trees and Linearly Ordered Sets, Handbook of Set Theoretic Topology, North Holland, 1984, 235-293 MR 776625 (86h:54040)

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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: https://doi.org/10.1090/S0002-9947-08-04503-0
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

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