Document Type: Research Paper
Authors
- O. A. Tantawy ^{1}
- S. A. El-Sheikh ^{2}
- R. A. Majeed ^{} ^{3}^{, 4}
^{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].
Keywords
- Smooth bitopological space
- supra smooth proximity
- smooth quasi-proximity
- compatibility
- FP-proximity map
Main Subjects
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