$T_2$ and $T_3$ objects at $p$ in the category of proximity spaces

Abstract

In previous papers, various notions of pre-Hausdorff, Hausdorff and regular objects at a point p in a topological category were introduced and compared. The main objective of this paper is to characterize each of these notions of pre-Hausdorff, Hausdorff and regular objects locally in the category of proximity spaces. Furthermore, the relationships that arise among the various PreT2, Ti, i = 0, 1, 2, 3, structures at a point p are investigated. Finally, we examine the relationships between the generalized separation properties and the separation properties at a point p in this category.