T3 and T4-Spaces in Smooth Topological Spaces

O. A. Tantawy; S. A. El-Sheikh; R. N. Majeed

doi:https://doi.org/10.14445/22315373/IJMTT-V13P510

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 13 | Number 1 | Year 2014 | Article Id. IJMTT-V13P510 | DOI : https://doi.org/10.14445/22315373/IJMTT-V13P510

T3 and T4-Spaces in Smooth Topological Spaces

O. A. Tantawy, S. A. El-Sheikh, R. N. Majeed

Abstract

In [25], we introduce the notion of r-fuzzy neighborhood filters in smooth topological spaces in sense of Gähler [10], and used it to define and study separation axioms Ti,i=0,1,2. Here we continue our study of the axioms of separation in smooth topological spaces. Therefore, we introduce the notion of r-fuzzy neighborhood filter at a set. Then by using this notion we define and study separation axioms Ti,i=3,4. These axioms are related only to usual points and ordinary subsets and reduce to axioms defined in [5], if τ: IX→{0,1}. So the current separation axioms are generalization of the old one. In addition, we show Ti-space not necessarily be a Ti-1-space for i=3,4. We give a condition for which, Ti-space is a Ti-1-space for i=3,4. Finally, these axioms are good extension from the point of view of Aygün et al. [2].

Keywords

Smooth topological space, Fuzzy filter, r-fuzzy neighborhood filter, Regular space, Normal space

References

[1] B. Amudhambigai, M. K. Uma and E. Roja, ‘‘Separation axioms in smooth fuzzy topological spaces,’’ Scientia Magna 5(3) (2009) 91-99.
[2] H. Aygün, M. W. Warner, and S. R. T. Kudri, ‘‘On smooth L-fuzzy topological spaces,’’ J. Fuzzy Math. 5(2) (1997) 321-338.
[3] R. Badard, ‘‘Smooth axiomatics,’’ First IFSA Congress, Palma de Mallorca (July 1986).
[4] F. Bayoumi and I. Ibedou, ‘‘Ti-spaces I,’’ J. Egypt. Math. Soc. 10(2) (2002) 179-199.
[5] F. Bayoumi and I. Ibedou, ‘‘Ti-spaces II,’’ J. Egypt. Math. Soc. 10 (2) (2002) 201-215.
[6] C. L. Chang, ‘‘Fuzzy topological spaces,’’ J. Math. Anal. Appl. 24 (1968) 182-190.
[7] K. C. Chattopadhyay, R. N. Hazra, and S. K. Samanta, ‘‘Gradation of openness: Fuzzy topology,’’ Fuzzy Sets and Systems 94 (1992) 237-242.
[8] K. C. Chattopadhyay and S. K. Samanta, ‘‘Fuzzy closure operator, fuzzy compactness and fuzzy connectedness,’’ Fuzzy Sets and Systems 54 (1993) 207-212.
[9] B. Daraby, ‘‘Regularity and Normality on L-Topological Spaces: (I),’’ General Mathematics 19(3) (2011) 67-73.
[10] P. Eklund and W. Gähler, ‘‘Fuzzy filter functors and convergence, in: Applications of category theory to fuzzy subsets,’’ Kluwer, Dordrecht/ Boston/ London (1992) 109-136.
[11] W. Gähler, ‘‘Convergence,’’ Seminarberichte aus dem Fachbereich der Fernuniversität Hagen 46 (1993) 31-73.
[12] W. Gähler, ‘‘The general fuzzy filter approch to fuzzy topology, I,’’ Fuzzy Sets and Systems 76 (1995) 205-224.
[13] W. Gähler, ‘‘The general fuzzy filter approch to fuzzy topology, II,’’ Fuzzy Sets and Systems 76 (1995) 225-246.
[14] W. Gähler, ‘‘Monadic convergence structures,’’ Siminarberichte aus dem Fachbereich Mathematik, Fren Universit t, Hagen, Band 67 (1999) 111-129.
[15] W. Gähler, ‘‘General topology - the monadic case, examples, applications,’’ Acta. Math. Hungar. 88(4) (2000) 279-290.
[16] B. Hutton, ‘‘Products of fuzzy topological spaces,’’ Topology Appl. 11 (1980) 50-67.
[17] Y. C. Kim and J. M. Ko, ‘‘Fuzzy G-closure operators,’’ Commun. Korean Math. Soc. 18(2) (2003) 325-340.
[18] Y. C. Kim, ‘‘Initial smooth fuzzy topological spaces,’’ J. of Korea Fuzzy Logic and Intelligent Systems 8(3) (1998) 88-94.
[19] R. Lowen, ‘‘Fuzzy topological spaces and fuzzy compactness,’’ J. Math. Anal. Appl. 56 (1976) 621-633.
[20] W. Peeters, ‘‘Subspaces of smooth fuzzy topologies and initial smooth fuzzy structures,’’ Fuzzy Sets and Systems 104(3) (1999) 423-433.
[21] A. A. Ramadan, ‘‘Smooth topological spaces,’’ Fuzzy Sets and Systems 48 (1992) 371-375.
[22] A. A. Ramadan, ‘‘Smooth topological spaces III,’’ J. Fuzzy Math. 8(1) (2000) 53-61.
[23] R. Srivastava, ‘‘On separation axioms in a newly defined fuzzy topology,’’ Fuzzy Sets and Systems, 62 (1994) 341-346.
[24] O. Tantawy, S. A. El-Sheikh, and R. Naser, ‘‘Fuzzy pairwise separation axioms in fuzzy bitopological spaces,’’ Jökull Journal 63(12) (2013) 243-260.
[25] O. A. Tantawy, S. A. El-Sheikh, and R. N. Majeed, ‘‘Lower separation axioms in smooth topological spaces,’’ submitted.
[26] S. Willard, ‘‘General topology,’’ Addison-Wesley Publishing Company (1970).
[27] Y. Yue and J. Fang, ‘‘On separation axioms in I-fuzzy topological spaces,’’ Fuzzy Sets and Systems 157 (2006) 780-793.

Citation :

O. A. Tantawy, S. A. El-Sheikh, R. N. Majeed, "T3 and T4-Spaces in Smooth Topological Spaces," International Journal of Mathematics Trends and Technology (IJMTT), vol. 13, no. 1, pp. 68-78, 2014. Crossref, https://doi.org/10.14445/22315373/IJMTT-V13P510