Theory of Generalized Compactness in Generalized Topological Spaces: Part I. Basic Properties

Abstract

In this paper, a novel class of generalized compact sets (briefly, g-Tg-compact sets ) in generalized topological spaces (briefly, Tg-spaces ) is studied. The study reveals that g-Tg -compactness implies ordinary compactness (briefly, Tg -compactness ) in Tg-spaces, and such statement implies its analogue in ordinary topological spaces (briefly, T -spaces ). Diagrams establish the various relationships amongst these types of g-Tg -compactness presented here and in relation to other types of g-T -compactness in T-spaces presented in the literature of Tg -spaces, and a nice application supports the overall theory.

Keywords

• Generalized topology
• generalized topological spaces
• generalized sets
• generalized compactness

References

1. Andrijević D., On b-open sets, Matematički Vesnik, 48, 59-64, 1996.
2. Arens R., Dugundji J., Remarks on the concept of compactness, Portuguese Mathematical Society (SPM), 9, 141 143, 1950.
3. 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.
4. Aruna C., Selvi R., τ∗ -Generalize b-Compactness and τ∗ -generalize b-connectedness in topological spaces, International Journal of Trend in Scientific Research and Development, 4(2), 2897-2900, 2018.
5. Balan K.B., Janaki C., On πp-compact spaces and πp-connectedness, International Journal of Scientific and Research Publications, 3(9), 1-3, 2013.
6. Bayhan S., Kanibir A., Reilly I.L., On functions between generalized topological spaces, Applied General Topology, 14(2), 195-203, 2013.
7. Benchalli S.S., Bansali P.M., gb-compactness and gb-connectedness in topological spaces, International Journal of Contemporary Mathematical Sciences, 6(10), 465-475, 2011.
8. Boonpok C., On generalized continuous maps in Čech closure spaces, General Mathematics, 19(3), 3-10, 2011.

9. Caldas M., Jafari S., On some applications of b-open sets in topological spaces, Memoirs of the Faculty of the Science, Kochi University Series A Mathematics, 2, 11-19, 2007.
10. Császár Á., γ -connected sets, Acta Mathematica Hungarica, 101(4), 273-279, 2003.
11. Császár Á., Generalized open sets in generalized topologies, Acta Mathematica Hungarica, 106(1-2), 53-66, 2005.
12. Császár Á., Remarks on quasi-topologies, Acta Mathematica Hungarica, 119(1-2), 197-200, 2008.
13. Dixmier J., General Topology, Springer-Verlag, 1984.
14. Dube K.K., Panwar O.S., Some properties of s -connectedness between sets and set s -connected mappings, Indian Journal of Pure and Applied Mathematics, 15(4), 343-354, 1984.
15. 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.
16. El-Monsef M.E.A., Radwan A.E., Ibrahem F.A., Nasir A.I., Some generalized forms of compactness, International Mathematical Forum, 7(56), 2767-2782, 2012.
17. Gál I.S., On the theory of (m, n) -compact topological spaces, Pacific Journal of Mathematics, 8(4), 721-734, 1958.
18. Greever J., On some generalized compactness properties, Publications of the Research Institute for Mathematical Sciences, Kyoto University Series A, 4, 39-49, 1968.
19. Jafari S., Noiri T., Properties of β -connected spaces, Acta Mathematica Hungarica, 101(3), 227-236, 2003.
20. 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.
21. John S.J., Thomas J., On soft μ-compact soft generalized topological spaces, Journal of Uncertainty in Mathematics Science, 2014(2014), 1-9, 2014.
22. Khodabocus M.I., Sookia N.-U.-H., Theory of generalized sets in generalized topological spaces, Journal of New Theory, 36, 18-38, 2021.
23. Khodabocus M.I., A Generalized Topological Spaces Endowed with Generalized Topologies, Ph.D. Thesis, University of Mauritius, 2020.
24. Krishnaveni K., Vigneshwaran M., bTμ -compactness and bTμ -connectedness in supra topological spaces, European Journal of Pure and Applied Mathematics, 10(2), 323-334, 2017.
25. Maheshwari S.N., Thakur S.S., On α-compact spaces, Bulletin of the Institute of Mathematics, Academia Sinica, 13, 341-347, 1985.
26. Mashhour A.S., Allam A.A., Mahmoud F.S., Khedr F.H., On supra topological spaces, Indian Journal of Pure and Applied Mathematics, 14(4), 502-510, 1983.
27. 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.
28. Njåstad O., On some classes of nearly open sets, Pacific Journal of Mathematics, 15(3), 961-970, 1965.
29. Pavlović V., Cvetković A.S., On generalized topologies arising from mappings, Matematički Vesnik, 38(3), 553-565, 2012.
30. Piękosz A., Wajch E., Compactness and compactifications in generalized topology, Topology and its Applications, 194, 241-268, 2015.
31. Robert A., Missier S.P., Connectedness and compactness via semi ∗α-open sets, International Journal of Mathematics Trends and Technology, 12(1), 1-7, 2014.
32. Sagiroglu S., Kanibir A., co−γ -compact generalized topologies and c -generalized continuous functions, Mathematica Balkanica, 23(1-2), 85-96, 2009.
33. Sekar S., Jothilakshmi B., On Semi-Generalized ∗ b-Connectedness and Semi-Generalized ∗ b-Compactness in Topological Spaces, Malaya Journal of Matematik, 3(1), 119-130, 2015.
34. Shi F.-G., Sβ -compactness in L-topological spaces, Proyecciones, 24(2), 153-165, 2005.
35. Singal M.K., Mathur A., On nearly compact spaces, Bollettino dell’Unione Matematica Italiana, 6(4), 702-710, 1969.
36. Solai R.S., g-compactness like in generalized topological spaces, Asia Journal of Mathematics, 1(2), 164-175, 2014.
37. Thomas J., John S.J., μ-compactness in generalized topological spaces, Journal of Advanced Studies in Topology, 3(3), 18-22, 2012.
38. Thomas J., John S.J., Soft generalized separation axioms in soft generalized topological spaces, International Journal of Scientific and Engineering Research, 6(3), 969-974, 2015.
39. Willard S., General Topology, Addison-Wesley Publishing Company, 1970.
40. 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.