Separable reduction theorems by the method of elementary submodels

TL;DR

This paper introduces a novel method using elementary submodels for proving separable reduction theorems, enabling the transfer of properties from nonseparable to separable spaces, with applications to various set and function properties.

Contribution

It develops a new approach based on elementary submodels for separable reduction, extending existing results to nonseparable spaces.

Findings

Method effectively reduces properties to separable subspaces

Applications include extending results of Zajicek, Lindenstrauss, Preiss

Enables analysis of dense, residual, and differentiability properties

Abstract

We introduce an interesting method of proving separable reduction theorems - the method of elementary submodels. We are studying whether it is true that a set (function) has given property if and only if it has this property with respect to a special separable subspace, dependent only on the given set (function). We are interested in properties of sets "to be dense, nowhere dense, meager, residual or porous" and in properties of functions "to be continuous, semicontinuous or Fr\'echet differentiable". Our method of creating separable subspaces enables us to combine our results, so we easily get separable reductions of function properties such as "be continuous on a dense subset", "be Fr\'echet differentiable on a residual subset", etc. Finally, we show some applications of presented separable reduction theorems and demonstrate that some results of Zajicek, Lindenstrauss and Preiss hold in nonseparable setting as well.