Post #3002041

I have in my mind two conflicting definitions of “f : X → Y has the Baire property (BP)”. (X and Y are topological spaces which I'm happy to assume Polish.) The first is that the preimage of an open subset has the BP (coincides with an open modulo a meager, and open can be replaced with Borel here). The second is that f is “Baire-measurable”, i.e., measurable with respect to the σ-algebras of BP subsets: the preimage of a BP has the BP. Did I dream up that these are equivalent? It comes down to showing that if the preimage of an open has the BP, then the preimage of a nowhere dense has the BP, but I'm stuck on that.