Baire reflection

Todorcevic, Stevo; Zoble, Stuart

doi:10.1090/S0002-9947-08-04503-0

Baire reflection
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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.

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