  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.
