New Types of μ -Proximity Spaces and Their Applications

Abstract

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 $\mu$ -clusters as the useful notion in the study of $\mu$ -proximity (or $\mu$ -nearness) spaces which recognize some of its features. Also, other types of $\mu$ -proximity, termed $R\mu$ -proximity and $O\mu$ -proximity, on $\mathcal{X}$ are defined. In a $\mu$ -proximity space $\left(\mathcal{X},{\delta }_{\mu }\right)$ , for any subset $K$ of $\mathcal{X}$ , one can find out nonempty collections ${\delta }_{\mu }\left[K\right]=\left\{G\subseteq \mathcal{X}\mid K{\overline{\delta }}_{\mu }G\right\}$ , which are hereditary classes on $\mathcal{X}$ . 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 $\mu$ -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 $\mu$ -proximities can be distinguished.