New Types of μ -Proximity Spaces and Their Applications

Near set theory supplies a major basis for the perception, differentiation, and classification of elements in classes that depend on their closeness, either spatially or descriptively. This study aims to introduce a lot of concepts; one of them is μ-clusters as the useful notion in the study of μ-proximity (or μ-nearness) spaces which recognize some of its features. Also, other types of μ-proximity, termed Rμ-proximity and Oμ-proximity, on X are defined. In a μ-proximity space X,δμ, for any subset K of X, one can find out nonempty collections δμK=G⫅X|Kδ¯μG, which are hereditary classes on X. Currently, descriptive near sets were presented as a tool of solving classification and pattern recognition problems emerging from disjoint sets; hence, a new approach to basic μ-proximity structures, which depend on the realization of the structures in the theory of hereditary classes, is introduced. Also, regarding to specific options of hereditary class operators, various kinds of μ-proximities can be distinguished.