Hyperspace approach to relation-based neighborhood operators

Abstract

Recently, new, subsequent versions of neighborhoods have been defined based on the concept of relation-based neighborhoods introduced by Y.Y. Yao. This article proposes a unified concept for investigations of such neighborhoods. This work presents the notion of hyper-neighborhood, which enables the investigation of the neighborhoods from the universe's perspective. As a result, we drive multiple equivalent characterizations of the types of neighborhoods that enable us to compare them and indicate the new, missing kinds of neighborhoods. Moreover, many kinds of neighborhoods defined in the literature on the issue proved to be identical. In particular, none of the types of recently defined so-called subset neighborhoods is new.