Set operators and associated functions

Year 2021, Volume: 70 Issue: 1, 456 - 467, 30.06.2021

Shyamapada Modak , Sk Selim

https://doi.org/10.31801/cfsuasmas.644689

Cited By: 2

Abstract

The study of two operators local function and the set operator $\psi$ on the ideal topological spaces are likely to be same to the study of closure and interior operator of the topological spaces. However, they are not exactly equal with the interior and closure operator of the topological spaces. In this context, we introduce two new set operators on the ideal topological spaces. Detail properties of these two operators are the part of this article. Furthermore, the operators interior (resp. $\psi$) and closure (local function) obey the relation $Int(A)$= X \ $Cl$(X \ A) (resp. $\psi$(A) = X \(X \A)$^*)$. We search the general method of these relations, through this manuscript.

Keywords

local function, associated function, $\psi_M$ operator, $\psi$ operator

References

• Al-Omari, A., Modak, S. and Noiri T., On θ-modifications of generalized topologies via hereditary classes, Commun. Korean Math. Soc. 31(4) (2016), 857-868. https://doi.org/10.4134/CKMS.c160002
• Bandhopadhya, C. and Modak, S., A new topology via Ψ-operator, Proc. Nat. Acad. Sci. India, Sect. A Phys. Sci. 76(A)(IV) (2006), 317-320.
• Dontchev, J., Idealization of Ganster-Reilly decomposition theorems, arXIV:math. Gn/9901017v1 [math.GN], 5 Jan 1999.
• Dontchev, J., Ganster, M. and Rose, D., Ideal resolvability, Topology Appl. 93 (1999), 1-16. https://doi.org/10.1016/S0166-8641(97)00257-5
• Hamlett, T. R. and Jankovic, D., Ideals in topological spaces and the set operator, Boll. Un. Mat.Ital. 4-B(7) (1990), 863-874.
• Hashimoto, H., On the *-topology and its applications, Fund. Math. 91 (1976), 5-10.
• Hayashi, E., Topologies defined by local properties, Math. Ann. 156 (1964), 205-215.
• Jankovic, D. and Hamlett, T. R., New topologies from old via ideals, Amer. Math. Monthly 97 (1990), 295-310, https://doi.org/10.1080/00029890.1990.11995593.
• Kuratowski, K., Topology, Vol. I, New York, Academic Press, 1966.
• Modak, S., Some new topologies on ideal topological spaces, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 82(3) (2012), 233-243. https://doi.org/10.1007/s40010-0120039-3
• Modak, S., Operators on grill M-space, Bol. Soc. Paran. Mat. 31(2) (2013), 101-107. https://doi.org/10.5269/bspm.v31i2.15547
• Modak, S., Grill-filter spacs, J.Indian Math. Soc. 80(3-4) (2013), 313-320.
• Modak, S., Minimal spaces with a mathematical structure, J. Assoc. Arab Univ. Basic Appl. Sci. 22 (2017), 98-101. https://doi.org/10.1016/j.jaubas.2016.05.005.
• Modak, S. and Bandyopadhyay, C., A note on Ψ-operator, Bull. Malays. Math. Sci. Soc. 30(1) (2007), 43-48.
• Modak, S., Garai, B. Mistry, S., Remarks on ideal M-space, Ann. Univ. Oradea Fasc. Mat. 19(1) (2012), 207-215.
• Modak, S. and Noiri, T., Connectedness of ideal topological spaces, Filomat 29(4) (2015), 661-665.https://doi.org/10.2298/FIL1504661M
• Newcomb, R. L., Topologies which are compact modulo an ideal, Ph. D. Dissertation, Univ. of Cal. at Santa Barbara (1967).
• Natkaniec, T., On I-continuity and I-semicontinuity points, Math. Slovaca 36(3) (1986), 297-312.
• Samuel, P., A topology formed from a given topology and ideal, J. London Math. Soc. 10 (1975), 409-416.
• Selim, Sk., Islam., Md. M. and Modak, S., Characterizations of Hayashi-Samuel Spaces via Boundary Points, Commun. Adv. Math. Sci. 2(3) (2019), 219-226. https://doi.org/10.33434/cams.546925
• Vaidyanathswamy, R., The localization theory in set-topology, Proc. Indian Acad. Sci. 20 (1945), 51-61. https://doi.org/10.1007/BF03048958