Smooth biproximity spaces and P-smooth quasi-proximity spaces

Subject Areas : History and biography

O. A. Tantawy ¹ , S. A. El-Sheikh ² , R. A. Majeed ³

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 | Department of Mathematics, Faculty of Education Abn Al-Haitham, Baghdad University, Baghdad, Iraq

Received: 2016-01-01 Accepted : 2017-08-20 Published : 2017-04-01

Keywords: compatibility, Smooth bitopological space, supra smooth proximity, smooth quasi-proximity, FP-proximity map,

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 relationshipbetween $(X,\delta_{12})$ and the $FP^*$-separation axioms which had been introduced by Ramadan etal. [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 $P$ smoothquasi-proximity spaces and prove that the associated smooth bitopological space $(X,\tau_\delta,\tau_{\delta^{-1}})$ satisfies $FP$-separation axioms in sense of Ramadan et al. [10].

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