Category product densities and liftings

In this paper we investigate two main problems. One of them is the question on the existence of category liftings in the product of two topological spaces. We prove, that if X × Y is a Baire space, then, given (strong) category liftings ρ and σ on X and Y, respectively, there exists a (strong) category lifting π on the product space such that π is a product of ρ and σ and satisfies the following section property: [ π ( E ) ] x = σ ( [ π ( E ) ] x ) for all  E ⊆ X × Y with Baire property and all x ∈ X . We give also an example, where some of the sections [ π ( E ) ] y must be without Baire property. we investigate the existence of densities respecting coordinates on products of topological spaces, provided these products are Baire spaces. The densities are defined on σ-algebras of sets with Baire property and select elements modulo the σ-ideal of all meager sets. In all the problems the situation in the “category case” turns out to be much better, than in case of products of measure spaces. In particular, in every product there exists a canonical strong density being a product of the canonical densities in the factors and there exist (strong) densities respecting coordinates with given a priori marginal (strong) densities.