In this paper we introduce the notation of shadowing sets which is a generalization of the notion of separating sets to the family of more than two sets. We prove that $${\bigcap_{i\in I}A_{i}}$$ is a shadowing set of the family $${\{A_{i}\}_{i\in I}}$$ if and only if $${\sum_{i\in I}A_{i}=\bigvee_{i\in I}\sum_{k\in I\setminus \{i\}}A_{i} + \bigcap_{i\in I}A_{i}}$$ . It generalizes the theorem stating that $${A\cap B}$$ is separating set for A and B if and only if $${A+B=A\cap B+A\vee B}$$ . In terms of shadowing sets, we give a criterion for an arbitrary upper exhauster to be an exhauster of sublinear function and a criterion for the minimality of finite upper exhausters. Finally we give an example of two different minimal upper exhausters of the same function, which answers a question posed by Vera Roshchina (J Convex Anal, to appear).