Document Type: Research Paper

**Authors**

Department of Mathematics, Annamalai University, Annamalai Nagar, Tamil Nadu 608 002, India

**Abstract**

We introduce a new class of fuzzy open sets called fuzzy $\bigwedge_e$ sets which includes the class of fuzzy $e$-open sets. We also define a weaker form of fuzzy $\bigwedge_e$ sets termed as fuzzy locally $\bigwedge_e$ sets. By means of these new sets, we present the notions of fuzzy $\bigwedge_e$ continuity and fuzzy locally $\bigwedge_e$ continuity which are weaker than fuzzy $e$-continuity and furthermore we investigate the relationships between these new types of continuity and some others.

**Keywords**

- fuzzy $bigwedge_e$ set
- fuzzy $bigvee_e$ set
- fuzzy locally $bigwedge_e$ set
- fuzzy $bigwedge_e$ continuity
- fuzzy locally $bigwedge_e$ continuity

**Main Subjects**

[1] K. K. Azad, On fuzzy semi continuity, fuzzy almost continuity and fuzzy weakly continuity, J. Math. Anal. Appl. 82 (1981), 14-32.

[2] F. G. Arenas, J. Dontchev, M. Ganster, On $lambda$ sets and the dual of generalized continuity, Questions and answers in General Topology. 15 (1997), 3-13.

[3] A. Bhattacharyya, M. N. Mukherjee, On fuzzy δ-almost continuous and δ ∗-almost continuous functions, J. Tripura Math. Soc. 2 (2000), 45-57.

[4] A. S. Bin Shahna, On fuzzy strong semi-continuity and fuzzy precontinuity, Fuzzy Sets and Systems. 44 (1991), 303-308.

[5] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968), 182-189.

[6] S. Debnath, On fuzzy δ semi continuous functions, Acta Cienc. Indica Math. 34 (2) (2008), 697-703.

[7] E. Ekici, On e-open sets, DP*-sets and DPE*sets and decomposition of coninuity, Arabian Journal for Science and Engineering. 33 (2A) (2008), 269-282.

[8] E. Ekici, On e ∗-open sets and (D, S) ∗ -sets, Mathematica Moravica. 13 (1) (2009), 29-36.

[9] E. Ekici, On a-open sets A∗-sets and decompositions of continuity and super-continuity, Annales Univ. Sci. Budapest. E¨otv¨os Sect. Math. 51 (2008), 39-51.

[10] E. Ekici, Some generalizations of almost contra-super-continuity, Filomat. 21 (2) (2007), 31-44.

[11] E. Ekici, New forms of contra-continuity, Carpathian Journal of Mathematics. 24 (1) (2008), 37-45.

[12] J. H. Park, B. Y. Lee, Fuzzy semi-preopen sets and fuzzy semi-precontinuous mappings, Fuzzy Sets and Systems. 67 (1994), 395-364.

[13] S. Ganguly, S. Saha, A note on semi-open sets in fuzzy topological spaces, Fuzzy Sets and Systems. 18 (1986), 83-96.

[14] P. P. Ming, L. Y. Ming, Fuzzy topology I, neighbourhood stucture of a fuzzy point and moore-smith convergence, J. Math. Anal. Appl. 76 (2) (1980), 571-599.

[15] H. Maki, Generalized sets and the associated closure operator, The special issue in commemoration of Professor Kazusada IKEDS Retirement, (1986), 139-146.

[16] M. N. Mukherjee, S. P. Sinha, On some weaker forms of fuzzy continuous and fuzzy open mappings on fuzzy topological spaces, Fuzzy Sets and Systems. 32 (1989), 103-114.

[17] A. Mukherjee, S. Debnath, δ-semi open sets in fuzzy setting, Journal Tri. Math. Soc. 8 (2006), 51-54.

[18] M. S. El Naschie, On the uncertainity of cantorian geometry and the two-slit experiment, Chaos, Solitons and Fractals. 9 (3) (1998), 517-529.

[19] M. S. El Naschie, On the unification of heterotic strings, M theory and e∞ theory, Chaos, Solitons and Fractals. 11 (14) (2000), 2397-2408.

[20] M. S. El Naschie, On a fuzzy Kahler-like manifold which is consistent with the two slit experiment, Int journal of Non-linear Sci Numer Simul. 6 (2005), 95-98.

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

[22] M. K. Singal, N. Prakash, Fuzzy Preopen sets and fuzzy preseparation axioms, Bull. Call. Math. Soc. 78 (1986), 57-69.

[23] V. Seenivasan, K. Kamala, Fuzzy e-continuity and fuzzy e-open sets, Annals of Fuzzy Mathematics and Informatics. 8 (1) (2014), 141-148.

[24] S. S. Thakur, S. Singh, On fuzzy semi-preopen sets and fuzzy semi-precontinuity, Fuzzy sets and systems. 98 (1998), 383-391.

[25] W. Shi, K. Liu, A fuzzy topology for computing the interior, boundary and exterior of spatial objects quantitatively in GIS, Comput Geosci. 33 (2007), 898-915.

[26] L. A. Zadeh, Fuzzy sets, Information Control. 8 (1965), 338-353.