In this paper, we use the notions of a minimal structure approximation space (short MSAS) and the notion of near open sets to introduce a new approximation of uncertain sets as a mathematical tool to modify the approximations. Relationships between these types are established via proof and counter examples. Also, some basic concepts of near approximations set are investigated and studied the relations between these different types of sets in MSAS. This set is a specific importance to help with the modifications of an approximation space via adding new concepts and facts. Finally, we use this concept to introduce the definitions of near lower approximation, near upper approximation, near boundary region, near rough and near exact sets and study some of the properties of this notion.

Pal and Bhattacharyya (1996)introduced the notion of pre--open sets.In this paper, we consider the class of pre--open sets in topological spaces and investigate some of their properties. Also,we present and study some weak separation axiomsby involving the notion of pre--open sets.