Convergence of the Associated Filters via Set-Operators

Abstract

                                                                                                                                                                                                                                                                                                             

Let (X, τ) be a topological space.     For a proper ideal I on (X, τ), the associated filter FI is defined and investigated in [1].    In this paper, we define several set-operators on an ideal topological space (X, τ, I) and investigate the relationship between the set-operators and limit points of the associated filter FI.

Keywords

• ideal
• Filter
• Associated filter
• Limit Point of a Filter
• Local Function

References

1. [1] Sk. Selim, T. Noiri, S. Modak, “Ideals and the associated filters on topological spaces,” Eurasian Bulletin of Mathematics, vol. 2(3), pp. 80-85, 2019.
2. [2] D. Janković, T. R. Hamlett, “New topologies from old via ideals,” The American Mathematical Monthly, vol. 97, pp. 295-310, 1990.
3. [3] T. Natkaniec, “On I-continuity and I-semicontinuity points,” Mathematica Slovaca, vol. 36(3), pp. 297-312, 1986.
4. [4] K. Kuratowski, Topology, Vol. I, New York, Academic Press, 1966.
5. [5] E. Hayashi, “Topologies defined by local properties,” Mathematische Annalen, vol. 156, pp. 205-215, 1964.
6. [6] H. Hashimoto, “On the  -topology and its applications,” Fundamenta Mathematicae, vol. 91, pp. 5-10, 1976.
7. [7] T. R. Hamlett, D. Janković, “Ideals in topological spaces and the set operator Ψ,” Bollettino dell'Unione Matematica Italiana., vol. 7, no. (4-B), pp. 863-874, 1990.
8. [8] S. Modak, “Some new topologies on ideal topological spaces,” Proceedings of the National Academy of Sciences, India, Sect. A Phys. Sci., vol. 82(3), pp. 233-243, 2012.

9. [9] P. Samuel, “A topology formed from a given topology and ideal,” Journal of the London Mathematical Society, vol.10, pp. 409-416, 1975.
10. [10] A. Al-Omari, H. Al-Saadi, “A topology via ω-local functions in ideal spaces,” Mathematica, vol. 60(83), pp. 103-110, 2018.
11. [11] C. Bandhopadhya, S. Modak, “A new topology via Ψ-operator,” Proceedings of the National Academy of Sciences, India, vol. 76(A), no. IV, pp. 317-320, 2006.
12. [12] S. Modak, C. Bandyopadhyay, “A note on Ψ-operator,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 30(1), 43-48, 2007. [13] K. D. Joshi, Introduction to General Topology, Michigan: Wiley, 1983.
13. [14] N. Bourbaki, General Topology, Chapter 1-4, Verlag, Berlin, Heidelberg: Springer, 1989.
14. [15] S. Modak, Sk. Selim and Md. M. Islam, “Sets and functions in terms of local function,” Submitted.