Smooth biproximity spaces and P-smooth quasi-proximity spaces

The notion of smooth biproximity space (X; 1; 2) where 1; 2 are gradation proximities dened by Ghanim et al. [10]. In this paper, we show every smooth biproximity space (X; 1; 2) induces a supra smooth proximity space 12 ner than 1 and 2. We study the relationship between (X; 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 T4, 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; 1 ) satises FP-separation axioms in sense of Ramadan et al. [10].