Proximal Relator Spaces


J. F. Peters




This article introduces proximal relator spaces. The basic approach is to define a nonvoid family of proximity relations $\mathcal R_{\delta_\Phi}$ (called a proximal relator) on a nonempty set. The pair $(X,\mathcal R_{\delta_\Phi})$ (also denoted $X(\mathcal R_{\delta_\Phi})$) is called a proximal relator space. Then, for example, the traditional closure of a subset of the Száz relator space $(X,\mathcal R)$ can be compared with the more recent descriptive closure of a subset of $(X,\mathcal R_{\delta_\Phi})$. 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 proximal relator space $(X,\mathcal R_{\delta_\Phi})$.