Smooth biproximity spaces and P-smooth quasi-proximity spaces

Journal of Linear and Topological Algebra

مقاله 1، دوره 06، شماره 02، تیر 2017، صفحه 91-107   

نوع مقاله: Research Paper

نویسندگان

O. A. Tantawy¹؛ S. A. El-Sheikh²؛ R. A. Majeed^{* 3، 4}

¹Department of Mathematics, Faculty of Science, Zagaziq University, Cairo, Egypt

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

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

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

چکیده

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 $(X,\tau_\delta,\tau_{\delta^{-1}})$ satisfies $FP$-separation axioms in sense of Ramadan et al. [10].

کلیدواژه‌ها

Smooth bitopological space؛ supra smooth proximity؛ smooth quasi-proximity؛ compatibility؛ FP-proximity map

مراجع

[1] S. E. Abbas, A study of smooth topological spaces, Ph.D. Thesis, South Vally University, Egypt, 2002.

[2] M. E. Abd El-Monsef, A. A. Ramadan, On fuzzy supratopological spaces, Indian Journal of Pure and Applied Mathematics. 18 (1987), 322-329.

[3] G. Artico, R. Moresco, Fuzzy proximities compatible with Lowen fuzzy uniformities, Fuzzy Sets and Systems. 21 (1987), 85-98.

[4] R. Badard, Smooth axiomatics, First IFSA Congress Palma de Mallorca, 1986.

[5] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968), 182-190.

[6] K. C. Chattopadhyay, R. N. Hazra, S. K. Samanta, Gradation of openness: Fuzzy topology, Fuzzy Sets and Systems. 94 (1992), 237-242.

[7] K. C. Chattopadhyay, S. K. Samanta, Fuzzy topology: Fuzzy closure operator, fuzzy compactness and fuzzy connectedness, Fuzzy Sets and Systems. 54 (1993), 207-212.

[8] V. A. Efremovic, Infinitesimal spaces, Doklady Akad. Nauk SSSR (N.S.) (in Russian) 76 (1951), 341-343.

[9] M. K. El Gayyar, E. E. Kerre and A.A. Ramadan, Almost compactness and near compactness in smooth topological spaces, Fuzzy Sets and Systems. 92 (1994), 193-202.

[10] M. H. Ghanim, O. A. Tantawy, F. M. Selim, Gradations of uniformity and gradations of proximity, Fuzzy sets and Systems. 79 (1996), 373-382.

[11] M. H. Ghanim, O. A. Tantawy, F. M. Selim, Gradation of supra-openness, Fuzzy Sets and Systems. 109 (2000), 245-250.

[12] A. Kandil, Biproximities and fuzzy bitopological spaces, Simon Stevin. 63 (1989), 45-66.

[13] A. Kandil, A. Nouh, S. A. El-Sheikh, On fuzzy bitopological spaces, Fuzzy Sets and Systemes. 74 (1995), 353-363.

[14] A. Kandil, A. A. Nouh, S. A. El-Sheikh, Fuzzy supra proximity spaces and fuzzy biproximity spaces, J. Fuzzy Math. 3 (1995), 301-315.

[15] A. K. Katsaras, Fuzzy proximity spaces, J. Math. Anal. Appl. 68 (1979), 100-110.

[16] A. K. Katsaras, On fuzzy proximity spaces, J. Math. Anal. Appl. 75 (1980), 671-583.

[17] Y. C. Kim, Mappings on fuzzy proximity and fuzzy uniform spaces, Kangweon-Kyungki Math. Jour. 4 (1996), 149-161.

[18] Y. C. Kim, J. W. Park, Some properties of fuzzy quasi-proximity spaces, Kangweon-Kyungki Math. Jour. 5 (1997), 35-49.

[19] Y. C. Kim, r-fuzzy semi-open sets in fuzzy bitopological spaces, Far East J. Math. Sci. special (FJMS) II. (2000), 221-236.

[20] E. P. Lee, Y.-B. Im, H. Han, Semiopen sets on smooth bitopological spaces, Far East J. Math. Sci. 3 (2001), 493-511.

[21] A. S. Mashhour, A. A. Allam, F. S. Mohmoud, F. H. Khedr, On supra topological spaces, Indian Journal of Pure and Applied Mathematics. 14 (1983), 502-510.

[22] A. A. Ramadan, Smooth topological spaces, Fuzzy Sets and Systems. 48 (1992), 371-375.

[23] A. A. Ramadan, S. E. Abbas, A. A. Abd El-Latif, On fuzzy bitopological spaces in Sostak’s sense, Commun. Korean Math. Soc. 21 (2006), 497-514.

[24] A. A. Ramadan, S. E. Abbas, A. A. El-Latif, On fuzzy bitopological spaces in Sostak’s sense (II), Commun.  Korean Math. Soc. 25 (2010), 457-475.

[25] S. K. Samanta, Fuzzy proximities and fuzzy uniformities, Fuzzy sets and Systems. 70 (1995), 97-105.

[26] A. P. Sostak, On a fuzzy topological structure, Suppl. Rend. Circ. Matem. Palermo, Ser.II. 11 (1985), 89-103.

[27] A. P. Sostak, Basic structures of fuzzy topology, J. Math. Sci. 78 (1996), 662-701.