Inscribing Closed Non-σ-lower Porous Sets into Suslin Non-σ-lower Porous Sets

This paper is a continuation of the work done in [9]. We are interested in the following question within the context of σ-ideals of σ-porous type. Let X be a metric space and let be a σ-ideal of subsets of X . Let S⊂ X be a Suslin set with S / ∈ . Does there exist a closed set F ⊂ S which is not in ? The answer is positive provided that X is locally compact and is a σ-ideal of σP-porous sets, where P is a porosity-like relation satisfying some additional conditions (see the definitions below, and for the precise statement, see [9]). In the case of the σideal of ordinary (i.e., upper) σ-porous sets, which satisfies the assumptions of the abovementioned theorem in any locally compact metric space, even more is true: X can be any topologically complete metric space (see [8]). The proofs are not easy; they use either some amount of descriptive set theory (see [9]) or a quite complicated construction (see [8]). In this paper, we deal with σ-ideals of σ-P-porous sets again, but these σ-ideals are supposed to be generated by closed P-porous sets, that is, every σ-P-porous set is covered by countably many closed P-porous sets. Note that this property does not hold for ordinary σ-porous sets but does hold for σ-lower porous sets. Although we will also work in nonseparable spaces, it turns out that the situation is much simpler than in [9]. Under a simple additional condition on the porosity-like relation P, we prove that every such σ-ideal has the property that every non-σ-P-porous Suslin subset of a topologically complete metric space X contains a closed non-σ-P-porous subset. As the main tool, we use a nonseparable version of Solecki’s theorem proved in [2].