ST2, ∆T2, ST3, ∆T3, Tychonoff, compact and ∂-connected objects in the category of proximity spaces

Abstract

In this paper, an explicit characterization of the separation properties $ST_2$, $\Delta T_2$, $ST_{3}$, $ \Delta T_{3} $ and Tychonoff objects are given in the topological category of proximity space. Furthermore, the (strongly) compact object and $\partial$-connected object are also characterized in the category of proximity space. Moreover, we investigate the relationships among $ST_2$, $\Delta T_2$, $ST_{3}$, $ \Delta T_{3} $, the separation properties at a point $p$, the generalized separation properties $T_{i}$, $i=0,1,2$, $\mathbf{T_{0}}$, $\mathbf{T_{1}}$, $\mathbf{T_{2}}$ and Tychonoff objects in this category. Finally, we investigate the relationships between $\partial$-connected object and (strongly) connected object in the topological category of proximity space.