Baire Reflection

We study reflection principles involving nonmeager sets and the Baire Property which are consequences of the generic supercompactness of ω2, such as the principle asserting that any point countable Baire space has a stationary set of closed subspaces of weight ω1 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 ω1 are meager in themselves. Unlike stronger forms of Game Reflection, these reflection principles do not decide CH, though they do give ω2 as an upper bound for the size of the continuum. Introduction and basic theory For a set A of ω-sequences of ordinals and a set of ordinals H the game G(A,H) has two players who alternate playing ordinals from H. Player II wins if the cooperative play belongs to A and loses otherwise. A weak version of the Game Reflection Principle defined and studied in [9], which we denote GRPω(θ) for an uncountable cardinal θ, asserts that for every A ⊂ θ, player II has a winning strategy in G(A, θ) if and only if II has a winning strategy in G(A,H) for an ω1club of H ∈ [θ]1 . Here an ω1-club means the set of H closed under some function f : θ → θ. The weakened reflection principle obtained by requiring each player to play finite sequences, rather than single ordinals (producing an element of θ by concatenation of the plays), is immediately equivalent to the Weak Baire Reflection Principle below. Definition 0.1. BRP(θ) asserts that any A ⊂ θ is meager if and only if A∩H is meager in H for an ω1-club of H ∈ [θ]1 . A stronger version requires reflection of a failure of the Baire Property. Definition 0.2. BRP (θ) asserts that any A ⊂ θ has the Baire Property in θ if and only if A ∩H has the Baire Property in H for an ω1-club of H ∈ [θ]1 . BRP and BRP will denote the global versions of these principles. Before deducing BRP from Game Reflection we discuss how to expand the scope of these principles to spaces other than the Generalized Baire Spaces (spaces of the form θ). Recall that the weight of a space X is the minimum cardinality of a base for Received by the editors March 10, 2006. 2000 Mathematics Subject Classification. Primary 03E55; Secondary 03E50.