Smooth biproximity spaces and P-smooth quasi-proximity spaces

The notion of smooth biproximity space  where $delta_1,delta_2$ are gradation proximities defined by Ghanim et al. [10]. In this paper, we show every smooth biproximity space $(X,delta_1,delta_2)$ induces a supra smooth proximity space $delta_{12}$ finer than $delta_1$ and $delta_2$. We study the relationship between $(X,delta_{12})$ and the $FP^*$-separation axioms which had been introduced by Ramadan et al. [23]. Furthermore, we show for each smooth bitopological space which is $FP^*T_4$, the associated supra smooth topological space is a smooth supra proximal. The notion of $FP$-(resp. $FP^*$) proximity map are also introduced. In addition, we introduce the concept of $P$ smooth quasi-proximity spaces and prove that the associated smooth bitopological space $(X,tau_delta,tau_{delta^{-1}})$ satises $FP$-separation axioms in sense of Ramadan et al. [10].