This is an announcement for the paper “Separable determination in Banach space” by Marek Cuthhttp://arxiv.org/find/math/1/au:+Cuth_M/0/1/0/all/0/1.

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 paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1608.03685