Moreover, we will prove that these two kinds of continuity produce different outcomes and broaden the notion of the G-method in topological spaces.
📌 3 Key Takeaways
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Objective & Hypothesis:In this paper we will discuss the similarities and differences for some topological properties in $G$-continuity and $G$-sequential continuity.
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Key Findings & Data:In particular we will illustrate the ways in which these notions agree and are divided in some topological properties by looking at their definitions, characteristics, and implications.
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Conclusion & Impact:Moreover, we will prove that these two kinds of continuity produce different outcomes and broaden the notion of the G-method in topological spaces.
In this paper we will discuss the similarities and differences for some topological properties in $G$-continuity and $G$-sequential continuity. In particular we will illustrate the ways in which these notions agree and are divided in some topological properties by looking at their definitions, characteristics, and implications. We will use examples from the literature \cite{counterexample} to distinguish between G-continuity and G-sequential continuity. Moreover, we will prove that these two kinds of continuity produce different outcomes and broaden the notion of the G-method in topological spaces.
Mucuk, O., Behram, S. (2026). About G-convergence and G-sequentially convergence. Proceedings of International Mathematical Sciences, Volume: 7 Issue: 2. https://doi.org/10.47086/pims.1598531
J. Connor, K.-G. Grosse-Erdmann, Sequential definitions of continuity for real functions, Rocky Mountain J. Math., 33(1) (2003), 93-121.
S. Lin, L. Liu, G-methods, G-spaces and G-continuity in topological spaces, Topology Appl., 212 (2016), 29-48.
H. Çakallı, Sequential definitions of compactness, Appl. Math. Lett., 21 (6) (2008), 594-598.
O. Mucuk, H. Çakallı, G-sequentially connectedness for topological groups with operations, Filomat, 32(3) (2018), 1079-1089.
Mucuk, O. and T. Şahan, On G-sequential Continuity, Filomat, 28 (6) (2014), 1181-1189.
S. Behram and O. Mucuk, About varieties of G-sequential methods, G-hulls and G-closures, Proceedings of International Mathematical Sciences, 5 (2) (2023), 81-86.
H. Çakallı, Sequential definitions of connectedness, Appl. Math. Lett., 25 (3) (2012), 461-465.
H. Çakallı, and O. Mucuk, On connectedness via a sequential method, Revista de la Unión Matemática Argentina, 54 (2) (2013), 101-109.
H. Çakallı, On G-continuity, Comput. Math. Appl., 61 (2011), 313-318.
O. Mucuk, S. Behram, Counter examples of G- convergent method, ICMS, AIP Conf. Proc. 2879, 070001 (2022).
Mucuk, S. Behram and H. Çakallı, G-convergently separation axioms, Filomat, 38 (18) (2024), 6433-6441.
S. Behram and O. Mucuk, G-convergence and G-sequential spaces, Hacet. J. Math. Stat. 54(6) (2025), 2326–2334.
L. Liu, S. Lin and X. Zhoua, On Gs-convergence and Gs-countable compactness, Topology Appl. 375 (2025), Paper No. 109554, 23 pp.
S. Behram. G-convergence and G-sequential spaces for G-methods, Ph.D Thesis, Erciyes University, (2025).
J. Connor, K.-G. Grosse-Erdmann, Sequential definitions of continuity for real functions, Rocky Mountain J. Math., 33(1) (2003), 93-121.
O. Mucuk, S. Behram, Counter examples of G- convergent method, ICMS, AIP Conf. Proc. 2879, 070001 (2022).