THEORY OF GENERALIZED CONNECTEDNESS (g-Tg-CONNECTEDNESS) IN GENERALIZED TOPOLOGICAL SPACES (Tg-SPACES)

Yıl 2023, Cilt: 6 Sayı: 1, 1 - 38, 31.01.2023

Mohammad Irshad Khodabocus , Noor-ul-hacq Sookıa

https://doi.org/10.33773/jum.1149387

Cited By: 2

Öz

In this paper, the definitions of novel classes of generalized connected sets (briefly, g-Tg-connected sets) and generalized disconnected sets
(briefly, g-Tg-disconnected sets) in generalized topological spaces (briefly, Tg-spaces) are defined in terms of generalized sets (briefly, g-Tg -sets) and, their
properties and characterizations with respect to set-theoretic relations are presented. The basic properties and characterizations of the notions of local, pathwise, local pathwise and simple g-Tg-connectedness are also presented. The study shows that local pathwise g-Tg -connectedness implies local g-Tg-connectedness, pathwise g-Tg-connectedness implies g-Tg-connectedness, and g-Tg-connectedness is a Tg-property. Diagrams establish the various relationships amongst these types of
g-Tg-connectedness presented here and in the literature, and a nice application supports the overall theory.

Anahtar Kelimeler

Generalized topological space (Tg-spaces), generalized local connectedness (local g-Tg-connectedness), generalized pathwise connectedness (pathwise g-Tg-connectedness), generalized local pathwise connectedness (local pathwise g-Tg-connectedness), generalized simple connectedness (simple g-Tg-connectedness), generalized components (g-Tg-components)

Kaynakça

• [1] A. V. Arhangel’skiĭ, R. Wiegandt, Connectedness and Disconnectedness in Topology, General Topology and its Applications, vol. 5, N. 1, pp. 9-33 (1975).
• [2] S. S. Benchalli, P. M. Bansali, gb-compactness and gb-connectedness Topological Spaces, International Journal of Contemporary Mathematical Sciences, vol. 6, N. 10, pp. 465-475 (2011).
• [3] R. X. Shen, A Note on Generalized Connectedness, Acta Mathematica Hungarica, vol. 122, N. 3, pp. 231-235 (2009).
• [4] J. Dixmier, General Topology, Springer Verlag New York Inc., vol. 18, pp. 31-35 (1997).
• [5] S. Willard, General Topology, Addison-Wesley Publishing Company, Reading, Massachusetts, vol. 18, pp. 31-35 (1970).
• [6] M. I. Khodabocus, N. -U. -H. Sookia, Theory of Generalized Compactness in Generalized Topological Spaces: Part II. Countable, Sequential and Local Properties, Fundamentals of Contemporary Mathematical Sciences, vol. 3, N. 2, pp. 98-118 (2022).
• [7] M. I. Khodabocus, N. -U. -H. Sookia, Theory of Generalized Compactness in Generalized Topological Spaces: Part I. Basic Properties, Fundamentals of Contemporary Mathematical Sciences, vol. 3, N. 1, pp. 26-45 (2022).
• [8] M. I. Khodabocus, N. -U. -H. Sookia, Theory of Generalized Separation Axioms in Generalized Topological Spaces, Journal of Universal Mathematics, vol. 5, N. 1, pp. 1-23 (2022).
• [9] M. I. Khodabocus, N. -U. -H. Sookia, Theory of Generalized Sets in Generalized Topological Spaces, Journal of New Theory, vol. 36, pp. 18-38 (2021).
• [10] M. I. Khodabocus, A Generalized Topological Space endowed with Generalized Topologies, PhD Dissertation, University of Mauritius, Réduit, Mauritius (2020).
• [11] A. A. El-Atik, H. M. A. Donia, A. S. Salama, On b-Connectedness and b-Disconnectedness and their Applications, Journal of the Egypt Mathematical Society, vol. 21, N. 1, pp. 63-67 (2013).
• [12] Á. Császár, γ-Connected Sets, Acta Mathematica Hungarica, vol. 101, N. 4, pp. 273-279 (2003).
• [13] S. Jafari, T. Noiri, Properties of β-Connected Spaces, Acta Mathematica Hungarica, vol. 101, N. 3, pp. 227-236 (2003).
• [14] O. Njåstad, On Some Classes of Nearly Open Sets, Pacific Journal of Mathematics, vol. 15, N. 3, pp. 961-970 (1965).
• [15] K. K Dube, O. S. Panwar, Some Properties of s-Connectedness between Sets and Set s-Connected Mappings, Indian Journal Pure and Applied Mathematics, vol. 15, N. 4, pp. 343-354 (1984).
• [16] A. Robert, S. P. Missier, Connectedness and Compactness via Semi∗ α-Open Sets, International Journal of Mathematics Trends and Technology, vol. 12, N. 1, pp. 1-7 (2014).
• [17] K. B. Balan, C. Janaki, On πp-Compact Spaces and πp-Connectedness, International Journal of Scientific and Research Publications, vol. 3, N. 9, pp. 1-3 (2013).
• [18] K. Krishnaveni, M. Vigneshwaran, bTµ -Compactness and bTµ -Connectedness in Supra Topological Spaces, European Journal of Pure and Applied Mathematics, vol. 10, N. 2, pp. 323-334 (2017).
• [19] F. M. V. Valenzuela, H. M. Rara, µ-rgb-Connectedness and µ-rgb-Sets in the Product Space in a Generalized Topological Space, Applied Mathematical Sciences Hikari Ltd, vol. 8, pp. 5261-5267 (2014).
• [20] I. Basdouri, R. Messaoud, A. Missaoui, Connected and Hyperconnected Generalized Topological Spaces, Proceedings of American Mathematical Society, vol. 5, N. 4, pp. 229-234 (2016).
• [21] C. Janaki, D. Sreeja, On πbµ -Compactness and πbµ -Connectedness in Generalized Topological Spaces, Journal of Academia and Industrial Research, vol. 3, N. 4, pp. 168-172 (2014).
• [22] W. K. Min, A Note on θ (g, g’) --Continuity in Generalized Topologies Functions, Acta Mathematica Hungarica, vol. 125, N. 4, pp. 387-393 (2009).
• [23] B. K. Tyagi, H. V. S. Chauhan, R. Choudhary, On γ-Connected Sets, International Journal of Computer Applications, vol. 113, N. 16, pp. 1-3 (2015).
• [24] A. Al-Omari, S. Modak, T. Noiri, On θ-Modifications of Generalized Topologies via Hereditary Classes, Communications of the Korean Mathematical Society, vol. 31, N. 4, pp. 857-868 (2016).
• [25] B. K. Tyagi, Harsh V. S. Chauhan, On Generalized Closed Sets in a Generalized Topological Spaces, CUBO A Mathematical Journal, vol. 18, N. 01, pp. 27-45 (2016).
• [26] Á. Császár, Remarks on Quasi-Topologies, Acta Mathematica Hungarica, vol. 119, N. 1-2, pp. 197-200 (2008).
• [27] Á. Császár, Generalized Open Sets in Generalized Topologies, Acta Mathematica Hungarica, vol. 106, N. 1-2, pp. 53-66 (2005).
• [28] V. Pavlović, A. S. Cvetković, On Generalized Topologies arising from Mappings, Matematički Vesnik, vol. 38, N. 3, pp. 553-565 (2012).
• [29] C. Boonpok, On Generalized Continuous Maps in Čech Closure Spaces, General Mathematics, vol. 19, N. 3, pp. 3-10 (2011).
• [30] A. S. Mashhour, I. A. Hasanein, S. N. E. Deeb, α-Continuous and α-Open Mappings, Acta Mathematica Hungarica, vol. 41, N. 3-4, pp. 213-218 (1983).
• [31] D. Andrijević, On b-Open Sets, Matematički Vesnik, vol. 48, pp. 59-64 (1996).
• [32] M. Caldas, S. Jafari, On Some Applications of b-Open Sets in Topological Spaces, Kochi Journal of Mathematics, vol. 2, pp. 11-19 (2007).