Types of Generalized $\delta$-Open Sets in Bitopological Spaces

Abstract

In theoretical and applied areas of mathematics, one can work with sets endowed with several structures. A bitopological space is a set equipped with two topologies. In this paper, some types of open sets weaker than delta-open sets are generalized to bitopological spaces and their corresponding interior and closure operators are introduced. The relations between these sets and  counter examples for the reverse relations are given. By using these sets, new types of continuous functions are defined and some of their properties are studied in bitopological spaces.

Keywords

• Bitopological spaces,$\delta$-open sets,preopen sets,semiopen sets

References

1. [1] D. Andrijevic, Some properties of the topology of $\alpha$-sets, Mat. Vesnik 36 (1984), 1-10.
2. [2] D. Andrijevic, Semi-pre-open sets, Mat. Vesnik 38 (1986), 24-32.
3. [3] A. A. Abo Khadra, A. A. Nasef, On extension of certain concepts from a topological space to a bitopological space, Proc. Math. Phys. Soc. Egypt 79 (2003), 91-102.
4. [4] E. Ekici, On $a$-open sets, $\mathcal{A}^*$-sets and decompositions of continuity and super-continuity, Ann. Univ. Sci. Budapest. E¨otv¨os Sect. Math. 51 (2008), 39-51.
5. [5] E. Ekici, On $e$-open sets, $\mathcal{DP}^*$-sets and $\mathcal{DPE}^*$-sets and decompositions of continuity, Arab. J. Sci. Eng. 33(2A) (2008), 269-282.
6. [6] E. Ekici, On $e^*$-open sets, $\mathcal{D,S}^*$-sets, Math. Morav. 13(1) (2009), 29-36.
7. [7] A. Ghareeb, T. Noiri, $\Lambda$-Generalized closed sets in bitopological spaces, J. Egyptian Math. Soc. 19 (2011), 142–145.
8. [8] A. Ghareeb, T. Noiri, $\Lambda$-Generalized closed sets with respect to an ideal bitopological space, Afr. Mat. 24 (2013), 97–101.

9. [9] H. Z. Ibrahim, $(p,q)$-$\beta$-$I$-$i$-open sets and $(p,q)$-$\beta$-$I$-$i$-almost continuous functions in ideal bitopological spaces, Univers. J. Appl. Math. 2(1) (2014), 63–71.
10. [10] M. İlkhan, M. Akyigit, E.E. Kara, On new types of sets via g-open sets in bitopological spaces, Commun. Fac. Sci. Univ. Ank. Series A1 67(1) (2018), 225–234.
11. [11] M. Jelic, A decomposition of pairwise continuity, J. Inst. Math. Comput. Sci. Math. Ser. 3 (1990), 641-656.
12. [12] J. C. Kelly, Bitopological spaces, J. Proc. London Math. Soc. 13 (1963), 71-89.
13. [13] F. H. Khedr, $C\alpha$-Continuity in bitopological spaces, Arab. J. Sci. Eng. 17(1) (1992), 85-89.
14. [14] F. H. Khedr, S. M. Al-Areefi, Precontinuity and semi-pre-continuity in bitopological spaces, Indian J. Pure Appl. Math. 23(9) (1992), 625-633.
15. [15] T. Noiri, V. Popa, Some properties of weakly open functions in bitopological spaces, Novi Sad J. Math. 36(1) (2006), 47–54.
16. [16] S. N. Maheshwari, R. Prasad, Semi open sets and semi continuous functions in bitopological spaces, Math. Notae. 26 (1977/78), 29-37.
17. [17] J. H. Park, B. Y. Lee, M. J. Son, On $\delta$-semiopen sets in topological space, J. Indian Acad. Math. 19(1) (1997), 59-67.
18. [18] S. Raychaudhuri, M. N. Mukherjee, On $\delta$-almost continuity and $\delta$-preopen sets, Bull. Inst. Math. Acad. Sin. 21 (1993), 357-366.
19. [19] G. Thamizharasi, P. Thangavelu, Remarks on closure and interior operators in bitopological spaces, J. Math. Sci. Comput. Appl. 1(1) (2010), 1–8.
20. [20] N. V. Veli c, H-closed topological spaces, Amer. Math. Soc. Transl. 78 (1968), 103-118.