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On separable determination of \(\sigma\)-\(\mathbf P\)-porous sets in Banach spaces. (English) Zbl 1327.46022

Summary: We use a method involving elementary submodels and a partial converse of Foran’s lemma to prove separable reduction theorems concerning Souslin \(\sigma\)-\(\mathbf P\)-porous sets where \(\mathbf P\) 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.

MSC:

46B26 Nonseparable Banach spaces
46G05 Derivatives of functions in infinite-dimensional spaces
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
54E35 Metric spaces, metrizability
58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
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