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

Abstract

We use a method involving elementary submodels and a partial converse of Foran lemma to prove separable reduction theorems concerning Souslin σ-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 Fréchet differentiable up to a cone small set.

Keywords

• Separable reduction
• Elementary submodel
• Foran system
• Porosity-like relation
• Cone small set