Document Type: Research Paper

**Author**

Department of Mathematics, Thiruvalluvar Arts and Science College (Affiliated to Thiruvalluvar University), Kurinjipadi, Tamil Nadu-607302, India.

**Abstract**

In this paper, the notion of generalized regular fuzzy irresolute, generalized regular fuzzy irresolute open and generalized regular fuzzy irresolute closed maps in fuzzy topological spaces are introduced and studied. Moreover, some separation axioms and $r$-GRF-separated sets are established. Also, the relations between generalized regular fuzzy continuous maps and generalized regular fuzzy irresolute maps are investigated. As a natural follow-up of the study of r-generalized regular fuzzy open sets, the concept of r-generalized regular fuzzy connectedness of a fuzzy set is introduced and studied.

**Keywords**

- Generalized regular fuzzy irresolute
- Generalized regular fuzzy irresolute open
- Generalized regular fuzzy irresolute closed mapping
- $r$-FRCO-$T_{1}$
- $r$-FRCO-$T_{2}$
- $r$-GRF-$T_{1}$
- $r$-GRF-$T_{2}$
- $r$-FRCO-regular
- $r$-FRCO-normal
- Strongly GRF-regular
- strongly GRF-normal
- $r$-GRF-separated sets
- $r$-GRF-connectedness

**Main Subjects**

[1] G. Balasubramanian and P. Sundaram, *On some generalizations of fuzzy continuous functions*, Fuzzy Set. Syst., 86 (1997), pp. 93-100.

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

[3] K.C. Chattopadhyay, R.N. Hazra, and S. K. Samanta, *Gradation of openness*, Fuzzy Set. Syst., 49 (1992), pp. 237-242.

[5] K.C. Chattopadhyay and S.K. Samanta, *Fuzzy topology, Fuzzy closure operator, Fuzzy compactness and Fuzzy connectedness*, Fuzzy Set. Syst., 54 (1993), pp. 207-212.

[5] E. Elavarasan, *On several types of generalized regular fuzzy continuous functions*, (Submitted).

[6] U. Hohle, *Upper semicontinuous fuzzy sets and applications*, J. Math. Anal. Appl., 78 (1980), pp. 659-673.

[7] U. Hohle and A.P. Sostak, *A general theory of fuzzy topological spaces*, Fuzzy Set. Syst., 73 (1995), pp. 131-149.

[8] U. Hohle and A.P. Sostak, *Axiomatic foundations of fixed-basis fuzzy topology*, The Hand-books of fuzzy sets series, 3, Kluwer academic publishers, Dordrecht (Chapter 3), (1999).

[9] Y.C. Kim and J.M. Ko, *$R$-generalized fuzzy closed sets*, J. Fuzzy Math., 12 (2004), pp. 7-21.

[10] T. Kubiak, *On fuzzy topologies*, Ph.D. Thesis, A. Mickiewicz, Poznan, (1985).

[11] T. Kubiak and A.P. Sostak, *Lower set-valued fuzzy topologies*, Quaest. Math., 20 (1997), pp. 423-429.

[12] S-J. Lee and E-P. Lee, *Fuzzy $r$-regular open sets and fuzzy almost $r$-continuous maps*, Bull. Korean Math. Soc., 39 (2002), pp. 441-453.

[13] N. Levine, *Generalized closed sets in topology*, Rend. Circ. Matem. Palermo, 19 (1970), pp. 89-96.

[14] A.A. Ramadan, S.E. Abbas, and Y.C. Kim, *Fuzzy irresolute mappings in smooth fuzzy topological spaces*, J. Fuzzy Math., 9 (2001), pp. 865-877.

[15] A.P. Sostak, *Two decades of fuzzy topology: Basic ideas, Notion and results*, Russian Math. Sur., 44 (1989), pp. 125-186.

[16] A.P. Sostak, *On a fuzzy topological structure*, Rend. Circ. Matem. Palermo Ser II, 11 (1986), pp. 89-103.

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

[18] A. Vadivel and E. Elavarasan, *Applications of $r$-generalized regular fuzzy closed sets*, Ann. Fuzzy Math. Infor., 12 (2016), pp. 719-738.