Separable determination in Banach spaces
We study a relation between three different formulations of theorems on separable determination: one using the concept of rich families, another via the concept of suitable models, and a third, 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, without using any terminology from logic or set theory.
Moreover, we prove that in Asplund spaces, generalized lushness is separably determined.