Tantawy, O., El-Sheikh, S., Majeed, R. (2017). Smooth biproximity spaces and P-smooth quasi-proximity spaces. Journal of Linear and Topological Algebra (JLTA), 06(02), 91-107.

O. A. Tantawy; S. A. El-Sheikh; R. A. Majeed. "Smooth biproximity spaces and P-smooth quasi-proximity spaces". Journal of Linear and Topological Algebra (JLTA), 06, 02, 2017, 91-107.

Tantawy, O., El-Sheikh, S., Majeed, R. (2017). 'Smooth biproximity spaces and P-smooth quasi-proximity spaces', Journal of Linear and Topological Algebra (JLTA), 06(02), pp. 91-107.

Tantawy, O., El-Sheikh, S., Majeed, R. Smooth biproximity spaces and P-smooth quasi-proximity spaces. Journal of Linear and Topological Algebra (JLTA), 2017; 06(02): 91-107.

Smooth biproximity spaces and P-smooth quasi-proximity spaces

^{1}Department of Mathematics, Faculty of Science, Zagaziq University, Cairo, Egypt

^{2}Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt

^{3}Department of Mathematics, Faculty of Science, Ain Shams University, Abbassia, Cairo, Egypt

^{4}Department of Mathematics, Faculty of Education Abn Al-Haitham, Baghdad University, Baghdad, Iraq

Abstract

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].

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