On a-star-I-Open Sets and a Decomposition of Continuity

Abstract

In this paper, we introduce a new set namely a$^{\ast }$-I-open set in ideal topological spaces. Besides, we give some properties and characterizations of it. We obtain that it is stronger than pre$^{\ast }$-$I$ -open set with b-open set and weaker than $\delta \beta _{I}$-open set. Finally, we give a decomposition of continuity by using a$^{\ast }$-I-open set as stated the following:" $f:\left( X,\tau ,I\right) \longrightarrow \left( Y,\varphi \right) $ is continuous if and only if it is a$^{\ast }$-$I$-continuous and strongly A$_{I}$-continuous."

Keywords

• a$^{\ast }$-\textit{I}-open set,Decomposition of continuity,Ideal

References

1. [1] M. E. Abd El-Monsef, S. N. El-Deeb and R. A. Mahmoud, $\beta $-open sets and $\beta $-continuous mapping, Bull. Fac. Sci. Assiut Univ. A12(1) (1983), 77-90.
2. [2] D. Andrijevi´c, On b-open sets, Mat. Vesnik, 48 (1996), 59-64.
3. [3] A. Caksu Güler and G. Aslim, b-I-open sets and decomposition of continuity via idealization, Proceedings of Institute of Mathematics and Mechanics, Natural Academy of Sciences of Azerbaijan, 22 (2005), 27-32.
4. [4] J. Dontchev, On pre-I-open sets and decompositon of I-continuity, Banyan Math. J., 2 (1996).
5. [5] E. Ekici and T. Noiri, On subsets and decompositions of continuity in ideal topological spaces, Arab. J. Sci. Eng. Sect. A Sci.,34 (2009), 165-167.
6. [6] A. A. El-Atik, A study on some types of mappings on topological spaces, MSc. Thesis, Tanta University, Egypt, 1997.
7. [7] E. Hatir and T. Noiri, On decompositions of continuity via idealization, Acta Math. Hungar., 96(4) (2002), 341-349.
8. [8] E. Hatir, A. Keskin and T. Noiri, A note on strong $\beta $-I-open sets and strongly $\beta$-I-continuous functions, Acta Math. Hungar., 108(1-2) (2005), 87-94.

9. [9] E. Hatir, A note on $\delta \alpha $-$I$-I-open sets and semi $semi^{\ast }$-$I$-I-open sets, Math. Commun., 16 (2011), 433-445.
10. [10] E. Hatir, On decompositions of continuity and complete continuity in ideal topological spaces, Europan Journal of Pure and Applied Math., 6(3) (2013), 352-362.
11. [11] D. Jankoviˇc and T.R. Hamlett, New topologies from old via ideals, Amer. Math. Monthly, 97 (1990), 295-310.
12. [12] K. Kuratowski, Topology, Vol. I, Academic Press, New York, 1966.
13. [13] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70 (1963), 36-41.
14. [14] A. S. Mashhour, M. E. Abd El-Monsef and S. N. El-Deeb, On precontinuous and weak precontinuous mappings, Proceedings of the Mathematical and Physical Society of Egypt, 53 (1982), 47-53.
15. [15] O. Njastad, On Some Classes of Nearly Open Sets, Pasific J. Math., 15 (1965), 961-970.
16. [16] R. Vaidyanathaswamy, The localisation theory in set topology, Proc. Indian Acad. Sci. Math. Sci., 20 (1945), 51-61.
17. [17] N. V. Veli´cko, H-closed topological spaces, Amer. Math. Soc. Transl., 78 (1968), 103-118.
18. [18] S. Yüksel, A. Acikgöz and T. Noiri, On delta-I-continuous functions, Turkish Journal of Mathematics, 29 (2005), 39-51.