Algebraic Separation and Shadowing of Arbitrary Sets

In this paper we consider a generalization of the separation technique proposed in [10,4,7] for the separation of finitely many compact convex sets $A_i,i\in I$ by another compact convex set $S$ in a locally convex vector space to arbitrary sets in real vector spaces. Then we investigate the notation of shadowing set which is a generalization of the notion of separating set and construct separating sets by means of a generalized Demyanov-difference in locally convex vector spaces.