Separable determination in Banach 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.

The content of Section 3 (concerning separable determination of generalized lushness) was previously contained in a preprint with the title "Separable determination of (generalized-)lushness". The paper "Separable determination of (generalized-)lushness" was withdrawn from arxiv, because it is not intended for publication as its content is covered here

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