The approximation number function and the characterization of covering approximation space

There is renewed interest in the topic of covering-based rough sets. It is important to understand the characterization of covering approximation spaces. Recently, Zhu et al. presented the concept of an approximation number function, which can be viewed as a quantitative tool for analyzing the covering approximation spaces. We focus on this concept and show that an approximation number function can be characterized by its properties. We give the axiomatic definition of an approximation number function, and prove that there is a one-to-one correspondence between the set of all lower approximation number functions and the set of all covering approximation spaces. This suggests that the concept is fundamental to the characterization of covering approximation spaces. We apply approximation number functions to the existing studies on approximation spaces, and a series of matroidal structures are induced in an approximation space.