G-convergence and G-sequential spaces

Abstract

In this paper, for a method $G$ on a set $X$, we consider $G$-convergence and $G$-sequential convergence; we analyse the distinction between them with some counterexamples. Then, $G$-sequential continuity is investigated, and the relations between $G$-continuous and $G$-sequentially continuous functions are determined. Moreover, $G$-sequential spaces are introduced and studied. Finally, examples of $G$-sequential spaces associated with $G$-methods are given.

Keywords

• G-method
• G-convergence
• G-sequential space

References

1. [1] A. Açıkgöz, H. Çakallı, F. Esenbel and L.D.R. Kocinac, A quest of G-continuity in neutrosophic spaces, Mathematical Methods in the Applied Sciences 44 (9), 7834- 7844, 2021.
2. [2] S. Behram and O. Mucuk, About varieties of G-sequential methods, G-hulls and Gclosures, Proceedings of International Mathematical Sciences 5(2), 81-86, 2023.
3. [3] J. Connor and K. G. Grosse-Erdmann, Sequential definitions of continuity for real functions, Rocky Mountain J. Math. 33 (1), 93-121, 2003.
4. [4] H. Çakallı, Sequential definitions of compactness, Appl. Math. Lett. 21 (6), 594-598, 2008.
5. [5] H. Çakallı, On G-continuity, Comput. Math. Appl. 61, 313-318, 2011.
6. [6] H. Çakallı, New kinds of continuities, Comput. Math. Appl. 61, 960-965, 2011.
7. [7] H. Çakallı, Sequential definitions of connectedness, Appl. Math. Lett. 25, 461-465, 2012.
8. [8] H. Çakallı and O. Mucuk, On connectedness via a sequential method, Revista de la Unión Matemática Argentina 54 (2), 101-109, 2013.

9. [9] M. Dik and I. Canak, New types of continuities, Abstr. Appl. Anal. 1-7, 2010.
10. [10] S. Lin and L. Liu, G-methods, G-sequentially spaces and G-continuity in topological spaces, Topology Appl. 212, 29-48, 2016.
11. [11] L. Liu, G-kernel-open sets, G-kernel-neighbourhoods and G-kernal-derived sets, J. Math. Res. Appl. 38 (3), 276-286, 2018.
12. [12] G. Di Maio and L.D.R. Kocinac, Statistical convergence in topology, Topology Appl. 156, 28-45, 2008.
13. [13] O. Mucuk, Topoloji and Kategory Teoreisi, 3rd Edition, Nobel Yayınları, Ankara, 2023. (in Turkish).
14. [14] O. Mucuk, and S. Behram, Some counter examples of G-convergence, AIP Conference Proceedings-Sixth International Conference of Mathematical Sciences, 2022.
15. [15] 0. Mucuk, S. Behram and H. Çakallı, G-convergently separation axioms, Filomat 38(18), 6433-6441, 2024.
16. [16] O. Mucuk and H. Çakallı, G-connectedness for topological groups with operations, Filomat 32 (3), 1079-1089, 2018.
17. [17] O. Mucuk and H. Çakall, On G-compactness of topological groups with operations, Filomat 36(20), 7113-7121, 2022.
18. [18] O. Mucuk and T. Sahan, On G-sequential continuity, Filomat 28 (6), 1181-1189, 2014.
19. [19] O. Mucuk and M.B. Teke, About sequentially open and closed subsets in product spaces, Maltepe Journal of Mathematics 2, 41-46, 2023.
20. [20] E. Öztürk, On almost-continuity and almost A-continuity of real functions, Commun. Fac. Sci. Univ. Ank. Seér. A1 Math. Stat. 32, 25-30, 1983.
21. [21] V.K. Srinivasan, An equivalent condition for the continuity of a function, Texas J. Sci. 32, 176-177, 1980.