smooth biproximity spaces and 𝑃-smooth quasi-proximity spaces

The notion of smooth biproximity space (𝑋, δ1, δ2) where δ1, δ2 are gradation proximities defined by ghanim et al. [10]. in this paper, we show every smooth biproximity space (𝑋, δ1, δ2) induces a supra smooth proximity space δ12 finer than δ1 and δ2. we study the relationship between (𝑋, δ12) 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, τδ, τδ−1 ) satisfies 𝐹𝑃-separation axioms in sense of ramadan et al. [10]