On separable determination of σ-P-porous sets in Banach spaces

We use a method involving elementary submodels and a partial converse of Foran lemma to prove separable reduction theorems concerning Souslin sigma-P-porous sets where P can be from a rather wide class of porosity-like relations in complete metric spaces. In particular, we separably reduce the notion of Souslin cone small set in Asplund spaces. As an application we prove that a continuous approximately convex function on an Asplund space is Frechet differentiable up to a cone small set. (C) 2014 Elsevier B.V. All rights reserved.