Rich families and elementary submodels

TL;DR

This paper compares two methods in functional analysis for proving separable reduction theorems, establishing their equivalence in certain spaces and applying this to projectional skeletons.

Contribution

It demonstrates the equivalence of rich families and elementary submodels methods in specific spaces and explores implications for projectional skeletons.

Findings

Results proved with rich families also hold with elementary submodels.

Equivalence is established in spaces with fundamental minimal systems and of density .

Application to projectional skeletons shows they can be indexed by ranges of projections.

Abstract

We compare two methods of proving separable reduction theorems in functional analysis -- the method of rich families and the method of elementary submodels. We show that any result proved using rich families holds also when formulated with elementary submodels and the converse is true in spaces with fundamental minimal system an in spaces of density $\aleph_1$. We do not know whether the converse is true in general. We apply our results to show that a projectional skeleton may be without loss of generality indexed by ranges of its projections.