Theory of Generalized Compactness in Generalized Topological Spaces: Part II. Countable, Sequential and Local Properties

Yıl 2022, Cilt: 3 Sayı: 2, 98 - 118, 28.07.2022

Mohammad Irshad Khodabocus , Noor-ul-hacq Sookıa

https://doi.org/10.54974/fcmathsci.1009492

Cited By: 3

Öz

In a recent paper, a novel class of generalized compact sets (briefly, g-Tg -compact sets) in generalized topological spaces (briefly, Tg -spaces) has been studied. In this paper, the concept is further studied and, other derived concepts called countable, sequential, and local generalized compactness (countable, sequential, local g-Tg -compactness) in Tg -spaces are also studied relatively. The study reveals that g-Tg -compactness implies local g-Tg -compactness and countable g-Tg-compactness, sequential g-Tg -compactness implies countable g-Tg -compactness and, g-Tg -compactness is a generalized topological property (briefly, Tg -property). Diagrams establish the various relationships amongst these types of g-Tg -compactness presented here and in the literature, and a nice application supports the overall theory.

Anahtar Kelimeler

Generalized topological spaces, generalized compactness, countable generalized compactness, sequential generalized compactness, local generalized compactness

Kaynakça

• Aruna C., Selvi R., On τ∗-generalize semi compactness and τ∗-generalize semi connectedness in topological spaces, International Journal of Scientific Research Engineering and Technology, 7(2), 74-78, 2018.
• Bacon P., The compactness of countably spaces, Pacific Journal of Mathematics, 32(3), 587-592, 1970.
• Butcher G.H., Joseph J.E., Characterizations of a generalized notion of compactness, Journal of the Australian Mathematical Society, 22(A), 380-382, 1976.
• Császár Á., Generalized open sets in generalized topologies, Acta Mathematica Hungarica, 106(1-2), 53-66, 2005.
• Duraiswamy I., View on compactness and connectedness via semi-generalised b-open sets, International Journal of Pure and Applied Mathematics, 118(17), 537-548, 2018.
• El-Monsef M.E.A., Kozae A.M., Some generalized forms of compactness and closedness, Delta Journal of Science, 9(2), 257-269, 1985.
• Greever J., On some generalized compactness properties, Publications of the Research Institute for Mathematical Sciences, Kyoto University Series A, 4, 39-49, 1968.
• Janaki C., Sreeja D., On πbμ -compactness and πbμ -connectedness in generalized topological spaces, Journal of Academia and Industrial Research, 3(4), 168-172, 2014.
• Khodabocus M.I., A Generalized Topological Spaces Endowed with Generalized Topologies, Ph.D. Thesis, University of Mauritius, 2020.
• Khodabocus M.I., Sookia N.-U.-H., Theory of generalized separation axioms in generalized topological spaces, Journal of Universal Mathematics, 5(1), 1-23, 2022.
• Khodabocus M.I., Sookia N.-U.-H., Theory of generalized compactness in generalized topological spaces; Part I. Basic properties, Fundamentals of Contemporary Mathematical Sciences, 3(1), 26-45, 2022.
• Khodabocus M.I., Sookia N.-U.-H., Theory of generalized sets in generalized topological spaces, Journal of New Theory, 36, 18-38, 2021.
• Maheshwari S.N., Thakur S.S., On α-compact spaces, Bulletin of the Institute of Mathematics, Academia Sinica, 13, 341-347, 1985.
• Mustafa J.M., μ-semi-compactness and μ-semi-Lindelöfness in generalized topological spaces, International Journal of Pure and Applied Mathematics, 78(4), 535-541, 2012.
• Piękosz A., Wajch E., Compactness and compactifications in generalized topology, Topology and its Applications, 194, 241-268, 2015.
• Sagiroglu S., Kanibir A., co-γ -compact generalized topologies and c -generalized continuous functions, Mathematica Balkanica, 23(1-2), 85-96, 2009.
• Solai R.S., g-compactness like in generalized topological spaces, Asia Journal of Mathematics, 1(2), 164-175, 2014.
• Thomas J., John S.J., μ-compactness in generalized topological spaces, Journal of Advanced Studies in Topology, 3(3), 18-22, 2012.
• Valenzuela F.M.V., Rara H.M., μ-rgb-connectedness and μ-rgb-sets in the product space in a generalized topological space, Applied Mathematical Sciences, 8(106), 5261-5267, 2014.