This article investigates extremality, stationarity, and regularity properties of infinite collections of sets in Banach spaces. Our approach strongly relies on the machinery developed for finite collections. When dealing with an infinite collection of sets, we examine the behaviour of its finite subcollections. This allows us to establish certain primal-dual relationships between the stationarity/regularity properties some of which can be interpreted as extensions of the Extremal principle. Stationarity criteria developed in the article are applied to proving intersection rules for Fr\'echet normals to infinite intersections of sets in Asplund spaces.
Published in Journal of Optimization Theory and Applications (2012) 154(2):339–369. The original publication is available at http://link.springer.com/article/10.1007%2Fs10957-012-0043-4