Separable determination in Banach spaces

TL;DR

This paper explores various formulations of separable determination in Banach spaces, establishing their equivalence and applying these results to properties like $\sigma$-porosity and generalized lushness, without relying on logic or set theory.

Contribution

It introduces a new formulation using $\omega$-monotone mappings and proves the equivalence of three approaches to separable determination in Banach spaces.

Findings

All formulations of separable determination are equivalent in Banach spaces.

New statements about $\sigma$-porosity are derived without set-theoretic terminology.

Generalized lushness is separably determined in Asplund spaces.

Abstract

We study a relation between three different formulations of theorems on separable determination - one using the concept of rich families, second via the concept of suitable models and third, a new one, suggested in this paper, using the notion of $\omega$-monotone mappings. In particular, we show that in Banach spaces all those formulations are in a sense equivalent and we give a positive answer to two questions of O. Kalenda and the author. Our results enable us to obtain new statements concerning separable determination of $\sigma$-porosity (and of similar notions) in the language of rich families; thus, not using any terminology from logic or set theory. Moreover, we prove that in Asplund spaces, generalized lushness is separably determined.