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We consider topological spaces X equipped with an algebra A of subsets of X and an ideal I of A . Motivated by the example of the Jordan measurable subsets of R , we consider the derived structure obtained by replacing A by the algebra ∂ A = { E ∈ A : ∂ E ∈ I } of sets with negligible boundaries, and I by ∂ I = I ∩ ∂ A . In a previous paper by M.R. Burke et al. (2012) [7] , the authors classified these derived structures (under some assumptions) and computed densities for them. In the present paper, we extend that work in the context of products of derived structures. We study in greater detail the box cross product γ ⊠ δ of two set maps γ ∈ P ( X ) A , δ ∈ P ( Y ) B introduced in joint work of the authors with K. Musial (2009) [6] , examining when it preserves densities and other types of liftings. For preservation of monotonicity, we introduce a variation on the localization property of ideals which is well-known for the meager ideal. An examination of skew products provides a class of structures to which our results apply.

We study the permanence of strong liftings for products of topological probability spaces. We get results to the positive for the 'usual product' or for the product σ-algebra obtained from it by adjoining the two-sided nil-null sets under assumptions implying that the product topology is contained in this product σ-algebras. For the construction of strong liftings in the product the application of marginals is basic as well as that of the fundamental properties of lifting topologies.

We extend earlier work [M.R. Burke, N.D. Macheras, K. Musial, W. Strauss, Category product densities and liftings, Topology Appl. 153 (2006) 1164–1191] of the authors on the existence of category liftings in the product of two topological spaces X and Y such that X × Y is a Baire space. For given densities ρ, σ on X and Y, respectively, we introduce two ‘Fubini type’ products ρ ⊙ σ and ρ ⊡ σ on X × Y . We present a necessary and sufficient condition for ρ ⊙ σ to be a density. Provided ( X , Y ) and ( Y , X ) have the Kuratowski–Ulam property, we prove for given category liftings ρ, σ on the factors the existence of a category lifting π on the product, dominating the density ρ ⊡ σ and such that π ( A × B ) = ρ ( A ) × σ ( B ) for Baire subsets A of X and B of Y , and ρ ( [ π ( E ) ] y ) = [ π ( E ) ] y for all y ∈ Y and Baire subsets E of X × Y . We show that further properties of consistency with the product structure cannot be expected. We prove also that contrary to measure theoretical liftings, in case of Baire spaces there might exist countably additive liftings. This answers, assuming the existence of a compact cardinal, a question from [M.R. Burke, N.D. Macheras, K. Musial, W. Strauss, Category product densities and liftings, Topology Appl. 153 (2006) 1164–1191]. The example we present is a version of an example of D.H. Fremlin of a space whose category algebra has a countably additive lifting.

Motivated by the construction of the algebra of Jordan measurable sets from the algebra of measurable sets in Euclidean spaces, we determine a natural class of structures ( X , S , A , I ) , where ( X , S ) is a topological space, A is an algebra of subsets of X , and I is an ideal of A , so that the derived structure ( X , S , ∂ A , ∂ I ) given by ∂ A : = { E ∈ A : ∂ E ∈ I } and ∂ I : = ∂ A ∩ I belongs to the same class. We provide a characterization of the derived structures whose ideal contains no nonempty open sets and derive from it that each such structure has a natural strong density and (assuming completeness) a strong lifting.

The purpose of this paper is to determine which metric spaces X and Y are such that the uniformly continuous maps f : X → Y are precisely the continuous maps between ( X , τ 1 ) and ( Y , τ 2 ) for some new topologies τ 1 and τ 2 on X and Y respectively.