# On fuzzy ??-sets and their applications.

## Keywords:

Fuzzy γ-open set, Fuzzy independent topology, FuzzyΛb-set, Fuzzy Λγ-set, Fuzzy Λb-continuity, Fuzzy Λγ-closed set## Abstract

The notion of ?-fuzzy set was introduced by M. E. EI-Shafei and A. Zakari in 2006 [20]. We examine some basic properties of it and prove some characterization theorems for the same. The paper presents a new class of fuzzy sets called fuzzy ??-sets that includes the class of all fuzzy ?-open sets. It also introduces the notion of fuzzy V?-sets as the dual concept of fuzzy ?? sets to study the spaces constituted by those sets and obtain a completely different structure which is called fuzzy independent Alexandorff space. A stronger form of fuzzy ?b - continuity [2]called fuzzy ??-continuity is introduced and the relationships are also established with the already existing functions accordingly. Finally, fuzzy ??-Generalized closed sets are defined and studied with some of their applications.## References

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[5] B. Bhattacharya and J. Chakraborty, Generalized regular fuzzy locally closed sets and some generalizations of fuzzy LC-continuous functions, Mathematical Sciences International Research Journal 3 (1), pp. 397-399, (2014).

[6] B.Bhattacharya,Fuzzy independent topological spaces generated by fuzzy ? -open sets and their application, accepted in Afrika Matematika, (2017).

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[10] A. Csaszar, ?-and ? -modification of generalized topologies, Acta Math. Hungar, 120, pp. 275-279, (2008).

[11] E. Ekici, On (LC, S)-continuous functions, Chaos, Solitons and Fractals, 38, pp. 430-438, (2001).

[12] S. M. Jafarian Amiri, A. Jafarzadeh and H. Khatibzade , An Alexandroff Topology on graphs, Bull. Of the Iranian Math. Soc., 39, pp. 647-662, (2013).

[13] A. Keskin and T. Noiri, Almost contra g-continuous function, Chaos, Solitons and Fractals, 42, pp. 238-246, (2013).

[14] M. S. El Naschie, On the certification of heterotic strings, theory and ? theory, Chaos, Solitons and Fractals, 2, pp. 397- 408, (2000).

[15] M. S. El Naschie, On a fuzzy Kahler-likl manifold which is consistent with the two slit experiment, Int. J. Nonlinear SciNumerSimul 205.

[16] G. Palani Chetty, Generalized Fuzzy Topology, Italian J. Pure Appl. Math., 24, pp. 91-96, (2008).

[17] A. Paul, B. Bhattacharya and J. Chakraborty, On ??- Set in Fuzzy Bitopological Spaces, Bol. Soc. Paran Mat. 35, n.3, pp. 285-299, (2017).

[18] A. Paul, B. Bhattacharya and J. Chakraborty, On ?-hyperconnectedness and fuzzy mappings in fuzzy bitopological spaces, Journal of Intelligent and Fuzzy Systems, 32, n. 3, pp. 1815-1820, (2017).

[19] M. K. Prakash Singal, Fuzzy pre-open sets and fuzzy pre separation axiom. Fuzzy Sets and system, 44, pp. 273-281, (1991).

[20] M. E. El Shafei and A. Zakari, ?-Generalized closed sets in fuzzy topological spaces, The Arabian Journal for Science and Engineering 31, N 24, (2006).

[21] S. S. Thakur and S. Singh, On fuzzy semi-pre-open sets and fuzzy semipro-continuity, Fuzzy Sets and system 98, pp. 383-391, (1998).

[22] G. Thangaraj and G. Balasubramanian, On Somewhat fuzzy continuous functions, J. Fuzzy Math. 11 (2), pp. 725- 736, (2003).

[23] C. K. Wong, Fuzzy points and local properties of fuzzy topology, Journal of Mathematical Analysis and Applications, 46, pp. 316-328, (1974).

[24] L. A. Zadeh, Fuzzy sets. Inform. And Control, 8, 338-353, (1965).

[25] B. S. Zhong, Fuzzy strongly semi-open set and fuzzy strongly semicontinuity, Fuzzy Sets and system, 52, pp. 345-351, (1992).

[2] G. Aslim and G. Gunel , On fuzzy ?b -sets and fuzzy ?b -continuity, Chaos, Solitons and Fractals, 42, pp. 1024-1030, (2009).

[3] K. K. Azad, Fuzzy semi continuity, fuzzy almost continuity and weakly continuitys. Math. Anal. Appl., 82, pp. 14-32, (1981).

[4] G. Balasubramanian and P. Sundaram, On some generalizations of fuzzy continuous functions, Fuzzy Sets and Systems, 86, pp. 93-100, (1997).

[5] B. Bhattacharya and J. Chakraborty, Generalized regular fuzzy locally closed sets and some generalizations of fuzzy LC-continuous functions, Mathematical Sciences International Research Journal 3 (1), pp. 397-399, (2014).

[6] B.Bhattacharya,Fuzzy independent topological spaces generated by fuzzy ? -open sets and their application, accepted in Afrika Matematika, (2017).

[7] C. L. Chang , Fuzzy topological spaces, J. Math. Anal. Appl., 24, pp. 338-353, (1968).

[8] A. Csaszar, Generalized topology, generalized continuity. Acta Math. Hungar, 96, pp. 351-357,(2002).

[9] A. Csaszar, Generalized open sets in generalized topologies, Acta Math. Hungar, 106, pp. 53-66, (2005).

[10] A. Csaszar, ?-and ? -modification of generalized topologies, Acta Math. Hungar, 120, pp. 275-279, (2008).

[11] E. Ekici, On (LC, S)-continuous functions, Chaos, Solitons and Fractals, 38, pp. 430-438, (2001).

[12] S. M. Jafarian Amiri, A. Jafarzadeh and H. Khatibzade , An Alexandroff Topology on graphs, Bull. Of the Iranian Math. Soc., 39, pp. 647-662, (2013).

[13] A. Keskin and T. Noiri, Almost contra g-continuous function, Chaos, Solitons and Fractals, 42, pp. 238-246, (2013).

[14] M. S. El Naschie, On the certification of heterotic strings, theory and ? theory, Chaos, Solitons and Fractals, 2, pp. 397- 408, (2000).

[15] M. S. El Naschie, On a fuzzy Kahler-likl manifold which is consistent with the two slit experiment, Int. J. Nonlinear SciNumerSimul 205.

[16] G. Palani Chetty, Generalized Fuzzy Topology, Italian J. Pure Appl. Math., 24, pp. 91-96, (2008).

[17] A. Paul, B. Bhattacharya and J. Chakraborty, On ??- Set in Fuzzy Bitopological Spaces, Bol. Soc. Paran Mat. 35, n.3, pp. 285-299, (2017).

[18] A. Paul, B. Bhattacharya and J. Chakraborty, On ?-hyperconnectedness and fuzzy mappings in fuzzy bitopological spaces, Journal of Intelligent and Fuzzy Systems, 32, n. 3, pp. 1815-1820, (2017).

[19] M. K. Prakash Singal, Fuzzy pre-open sets and fuzzy pre separation axiom. Fuzzy Sets and system, 44, pp. 273-281, (1991).

[20] M. E. El Shafei and A. Zakari, ?-Generalized closed sets in fuzzy topological spaces, The Arabian Journal for Science and Engineering 31, N 24, (2006).

[21] S. S. Thakur and S. Singh, On fuzzy semi-pre-open sets and fuzzy semipro-continuity, Fuzzy Sets and system 98, pp. 383-391, (1998).

[22] G. Thangaraj and G. Balasubramanian, On Somewhat fuzzy continuous functions, J. Fuzzy Math. 11 (2), pp. 725- 736, (2003).

[23] C. K. Wong, Fuzzy points and local properties of fuzzy topology, Journal of Mathematical Analysis and Applications, 46, pp. 316-328, (1974).

[24] L. A. Zadeh, Fuzzy sets. Inform. And Control, 8, 338-353, (1965).

[25] B. S. Zhong, Fuzzy strongly semi-open set and fuzzy strongly semicontinuity, Fuzzy Sets and system, 52, pp. 345-351, (1992).

## Published

2019-05-30

## How to Cite

[1]

B. Bhattacharya, A. Paul, and J. Chakraborty, “On fuzzy ??-sets and their applications.”,

*Proyecciones (Antofagasta, On line)*, vol. 38, no. 2, pp. 237-253, May 2019.## Issue

## Section

Artículos