## Baire reflection

HTML articles powered by AMS MathViewer

- by Stevo Todorcevic and Stuart Zoble PDF
- Trans. Amer. Math. Soc.
**360**(2008), 6181-6195 Request permission

## 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.## References

- Cummings, J.,
*Large Cardinal Properties of Small Cardinals*, 1988, unpublished manuscript - Ryszard Engelking,
*General topology*, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR**1039321** - Qi Feng and Thomas Jech,
*Local clubs, reflection, and preserving stationary sets*, Proc. London Math. Soc. (3)**58**(1989), no. 2, 237–257. MR**977476**, DOI 10.1112/plms/s3-58.2.237 - Qi Feng, Menachem Magidor, and Hugh Woodin,
*Universally Baire sets of reals*, Set theory of the continuum (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 26, Springer, New York, 1992, pp. 203–242. MR**1233821**, DOI 10.1007/978-1-4613-9754-0_{1}5 - Matthew Foreman and Stevo Todorcevic,
*A new Löwenheim-Skolem theorem*, Trans. Amer. Math. Soc.**357**(2005), no. 5, 1693–1715. MR**2115072**, DOI 10.1090/S0002-9947-04-03445-2 - D. H. Fremlin,
*Consequences of Martin’s axiom*, Cambridge Tracts in Mathematics, vol. 84, Cambridge University Press, Cambridge, 1984. MR**780933**, DOI 10.1017/CBO9780511896972 - F. Galvin, T. Jech, and M. Magidor,
*An ideal game*, J. Symbolic Logic**43**(1978), no. 2, 284–292. MR**485391**, DOI 10.2307/2272827 - T. Jech, M. Magidor, W. Mitchell, and K. Prikry,
*Precipitous ideals*, J. Symbolic Logic**45**(1980), no. 1, 1–8. MR**560220**, DOI 10.2307/2273349 - Bernhard König,
*Generic compactness reformulated*, Arch. Math. Logic**43**(2004), no. 3, 311–326. MR**2052885**, DOI 10.1007/s00153-003-0211-1 - Paul Larson,
*Separating stationary reflection principles*, J. Symbolic Logic**65**(2000), no. 1, 247–258. MR**1782117**, DOI 10.2307/2586534 - William Mitchell,
*Aronszajn trees and the independence of the transfer property*, Ann. Math. Logic**5**(1972/73), 21–46. MR**313057**, DOI 10.1016/0003-4843(72)90017-4 - Saharon Shelah,
*Cardinal arithmetic*, Oxford Logic Guides, vol. 29, The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications. MR**1318912** - John R. Steel,
*PFA implies $\textrm {AD}^{L(\Bbb R)}$*, J. Symbolic Logic**70**(2005), no. 4, 1255–1296. MR**2194247**, DOI 10.2178/jsl/1129642125 - Alan D. Taylor,
*Regularity properties of ideals and ultrafilters*, Ann. Math. Logic**16**(1979), no. 1, 33–55. MR**530430**, DOI 10.1016/0003-4843(79)90015-9 - Stevo Todorčević,
*Localized reflection and fragments of PFA*, Set theory (Piscataway, NJ, 1999) DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 58, Amer. Math. Soc., Providence, RI, 2002, pp. 135–148. MR**1903856**, DOI 10.4310/mrl.2002.v9.n4.a6 - S. Todorčević,
*Trees and linearly ordered sets*, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 235–293. MR**776625**

## 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
- MR Author ID: 172980
- 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
- Received by editor(s): March 10, 2006
- Published electronically: July 24, 2008
- © Copyright 2008 American Mathematical Society
- 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
- MathSciNet review: 2434283