Proximal Relator Spaces

Abstract

This article introduces proximal relator spaces.  The basic approach is to define a nonvoid family of proximity relations $\mathcal{R}_{\delta}$ (called a proximal relator) on a nonempty set.  The pair $(X,\mathcal{R}_{\delta})$ (also denoted $X(\mathcal{R}_{\delta})$) is called a proximal relator.  Then, for example, the traditional closure of a subset of the Sz\'{a}z relator space $X(\mathcal{R})$ can be compared with the more recent descriptive closure of a subset of $X(\mathcal{R}_{\delta})$.  This leads to an extension of fat and dense subsets of the relator space $X(\mathcal{R})$ to proximal fat and dense subsets of the space $\mathcal{R}_{\delta}$.