Study of Maximal Open Sets and Its Image with Ideals

Chhapikul Miah; Shyamapada Modak; Md. Monirul  Islam

doi:10.5269/bspm.65362

Chhapikul Miah Sukanta Mahavidyalaya
Shyamapada Modak University of Gour Banga
Md. Monirul Islam Kaliachak College

Abstract

The operator $\psi$ has been introduced as an associated set-valued set-function. But it is also important for the study of maximal open sets. As a result of these studies, maximal $I$-open sets and maximal $I^{*}$-open sets have been introduced through this write up. Several characterizations of maximal $I$-open sets and maximal $I^{*}$-open sets have been studied in this paper. The role of maximal $I$-open sets and maximal $I^{*}$-open sets in the cofinite subsets has also been discussed in this paper. In aspects of topological invariant, the homeomorphic images of maximal $I$-open set and maximal $I^{*}$-open set have been discussed here.

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References

1. Abd El-Monsef M.E., Lashien E. F., Nasef A. A., {\em On $I$-open sets and $I$-continuous functions}, Kyung-pook Math. J., 32(1), 21-30, (1992).
2. Hamlett, T. R., Jankovi\'{c}, D., {\em Ideals in topological spaces and set operator $\psi$}, Bull. U.M.I., 7(4-B), 863-874, (1990).
3. Jankovi\'{c} D. and Hamlett T.R., {\em New topologies from old via ideals}, Amer. Math. Monthly, 97(4), 295-300, (1990).
4. Jankovi\'{c} D., {\em Compatible extensions of Ideals}, Boll.U.M.I., 7, 453-465, (1992).
5. Hoque J., Modak S. and Acharjee S., {\em Filter versus ideal on the topological spaces}, Accepted for publication book chapter.
6. Kuratowski K., {\em Topology}, Vol.1 Academic Press, New York, (1966).
7. Mashhour A. S., Hasancin I. A., EI-Deeb S.N., {\em A note on semi-continuity and precontinuity}, Indian J. Pure Appl. Math., 13(10), 1119-1123, (1982).
8. Modak S. and Bandhyopadhyay C., {\em A note on $\psi$-operator}, Bull. Malyas. Math. Sci. Soc., 30(1), 43-48, (2007).
9. Modak S., Miah C. and Md. M.Islam, {\em Role of $\psi$-operator in the study of minimal $I$-open sets}, Iraqi. J. Sci., Communicated.
10. Modak S. and Selim Sk., {\em Set operators and associated functions}, Commun. Fac. Sci. Univ. A1 Math. Stat., 70(1), 456-467, (2021).
11. Nakaoka F. and Oda N., {\em Some applications of maximal open sets}, Int. J. Math. Math. Sci., 23, 1331-1340, (2003).
12. Natkaniec T., {\em On \textbf{I}-continuity and \textbf{I}-semicontinuity points}, Math. Slovaca, 36(3), 297-312, (1986).
13. Newcomb R.L., {\em Topologies which are compact modulo an ideal}, Ph.D.Dissertation, Univ. of Cal. at Santa Barbara, (1967).
14. Nj{\aa}stad O., {\em On some classes of nearly open sets}, Pacific J.Math., 15(3), 961-970, (1965).
15. Nj{\aa}stad O., {\em Remarks on topologies defined by local properties}, Norske Vid. Akad. Oslo Mat.-Nature. Kl. Skr. (N. S.), 8, 1-16, (1966).
16. N.R.Pach\'on Rubiano, {\em Between closed and $I_{g}$-closed sets}, Eur.J.Pure Aplied Math., 11(1), 200-314, (2018).
17. Rashid N. O. and Hussein S. A., {\em Maximal and Minimal Regular $\beta$-Open Sets in Topological Spaces}, Iraqi. J. Sci., 63(4) , 1720-1728, (2022).
18. Selim Sk., Noiri T. and Modak S., {\em Ideals and the associated filters on topological spaces}, Eurasian Bulletin of Mathematics, 2(3), 80-85, (2019).
19. Vaidyanathaswamy R., {\em Set topology}, Chelsea Publishing Company, (1960).