  This paper continues studies of non-intersection properties of finite
collections of sets initiated 40 years ago by the extremal principle. We study
elementary non-intersection properties of collections of sets, making the core
of the conventional definitions of extremality and stationarity. In the setting
of general Banach/Asplund spaces, we establish new primal (slope) and dual
(generalized separation) necessary conditions for these non-intersection
properties. The results are applied to convergence analysis of alternating
projections.
