Al-Omeri, Wadei; Abu-Saleem, M. \(\mathcal{I}_\mathbf{g}^*\)-closed sets via ideal topological spaces. (English) Zbl 1452.54001 Missouri J. Math. Sci. 31, No. 2, 174-191 (2019). Summary: In this paper, aspects of generalized continuity and generalized closedness are explored. The standard material on the notions of \(*g\)-open, \(\mathbf{g}\)-open sets and some definitions and results that are needed are presented first. Then the class of \(\mathcal{I}_\mathbf{g}^*\)-closed sets is introduced and its fundamental properties are studied. Also, \(\mathcal{I}_\mathbf{g}^*\)-regular, \(^*\)-additive, \(^*\)-multiplicative, \(\mathcal{I}_\mathbf{g}^*\)-additive, and \(\mathcal{I}_\mathbf{g}^*\)-multiplicative spaces are introduced and their properties are investigated. Cited in 1 Document MSC: 54A05 Topological spaces and generalizations (closure spaces, etc.) 54C08 Weak and generalized continuity Keywords:generalized closed; \(g\)-closed set; semi-open; pre-open; ideal topological space; \(^*\)-additive; \(^*\)-multiplicative; \( \mathcal{I}_\mathbf{g}^*\)-closed; \( \mathcal{I}_\mathbf{g}^*\)-connected; \( \mathcal{I}_\mathbf{g}^*\)-regular spaces; semi\(^*\)-\(\mathcal{I} \)-open × Cite Format Result Cite Review PDF Full Text: DOI Euclid References: [1] M. E. Abd El-Monsef, E. F. Lashien, and A. A. Nasef, Some topological operations via ideals, Kyungpook Mathematical Journal, 32.2 (1992), 273-284. · Zbl 0874.54002 [2] W. Al-Omeri, M. Noorani, A. Al-Omari, and T. Noiri, Weak separation axioms via e-\(I\)-sets in ideal topological spaces, Eur. J. Pure Appl. Math., 8.4 (2015), 502-513. · Zbl 1420.54002 [3] W. Al-Omeri, M. S. M. Noorani, and A. Al-Omari, New forms of contra-continuity in ideal topology spaces, Missouri J. Math. Sci., 26.1 (2014), 33-47. · Zbl 1300.54005 · doi:10.35834/mjms/1404997107 [4] W. Al-Omeri, M. S. M. Noorani, and A. Al-Omari, Weak open sets on simple extension ideal topological space, Italian Journal of Pure and Applied Mathematics, 33 (2014), 333-344. · Zbl 1330.54002 [5] W. Al-Omeri, M. Noorani, A. Al-Omari, On e-\(I\)-open sets, e-\(I\)-continuous functions and decomposition of continuity, J. Math. Appl., 38 (2015), 15-31. · Zbl 1382.54005 [6] W. Al-Omeri and T. Noiri, \( \mathcal{A}\mathcal{G}_{\I^*} \)-sets, \( \mathcal{B}G_{\I^*} \)-sets and \(\delta\beta_I\)-open sets in ideal topological spaces, International Journal of Advances in Mathematics, 2018.4 (2018), 25-33. [7] F. G. Arenas, J. Dontchev, and M. L. Puertas, Idealization of some weak separation axioms, Acta Math. Hungar, 89.1-2 (2000), 47-53. · Zbl 0958.54020 [8] S. P. Arya and T. M. Nour, Characterizations of s-normal spaces, Indian Journal of Pure and Applied Mathematics, 21.8 (1990a), 717-719. · Zbl 0706.54021 [9] P. Bhattacharya and B. K. Lahiri, Semi-generalized closed sets in topology, Indian Journal of Mathematics, 29 (2011), 375-382. · Zbl 0687.54002 [10] V. R. Devi, D. Sivaraj, and T. T. Chelvam, Codense and completely codense ideals, Acta Mathematica Hungarica, 108 (2005), 197-205. · Zbl 1100.54001 · doi:10.1007/s10474-005-0220-0 [11] J. Dontchev, On generalizing semi-preopen sets, Memoirs of the Faculty of Science Kochi University Series A Mathematics, 16 (1995), 35-48. · Zbl 0833.54001 [12] J. Dontchev, M. Ganster, and T. Noiri, Unified approach of generalized closed sets via topological ideals, Mathematics Japonica, 49 (1999a), 395-401. · Zbl 0991.54043 [13] J. Dontchev, On some separation axioms associated with \(\alpha \)-topology, Memoirs of the Faculty of Science Kochi University Series A Mathematics, 18 (1997), 31-35. · Zbl 0878.54009 [14] Y. Gnanambal, On generalized preregular closed sets in topological spaces, Indian Journal of Pure and Applied Mathematics, 28.3 (1997), 351-360. · Zbl 0942.54003 [15] E. Hatir and T. Noiri, On decompositions of continuity via idealization, Acta Math. Hungar., 96 (2002), 341-349. · Zbl 1012.54019 · doi:10.1023/A:1019760901169 [16] E. Hatir and S. Jafari, On weakly semi-i-open sets and another decomposition of continuity via ideals, Sarajevo J. Math., 2 no. 14 (2006), 107-114. · Zbl 1123.54301 [17] E. Hatir and T. Noiri, Weakly pre-i-open sets and decomposition of continuity, Acta Mathematica Hungarica, 106 no. 3 (2005), 227-238. · Zbl 1081.54003 · doi:10.1007/s10474-005-0015-3 [18] P. M. Helen, P. Selvarani, and V. Vijayan, \(g^{*S}I\)-closed in ideal topological spaces, International Journal of Mathematical Archive, 3 no. 6 (2012), 2402-2415. [19] D. Jankovic and T. R. Hamlett, New topologies from old via ideals, Amer. Math. Monthly, 97 (1990), 295-310. · Zbl 0723.54005 · doi:10.1080/00029890.1990.11995593 [20] A. Kar and P. Bhattacharyya, Weakly semi-continuous functions, J. Indian Acad. Math., 8 (1986), 83-93. · Zbl 0624.54010 [21] A. Kar and P. Bhattacharyya, Weakly semi-continuous functions, J. Indian Acad. Math., 8 (1986), 83-93. · Zbl 0624.54010 [22] A. Kar and P. Bhattacharyya, Weakly semi-continuous functions, J. Indian Acad. Math., 8 (1986), 83-93. · Zbl 0624.54010 [23] E. D. Khalimsky, Applications of connected ordered topological spaces in topology, Conference of Math. Department of Povolsia, 1970. [24] M. Khan and T. Noiri, On gI-closed sets in ideal topological spaces, Journal of Advanced Studies in Topology, 1 (2010a), 29-33. · Zbl 1196.54001 · doi:10.20454/jast.2010.202 [25] M. Khan and T. Noiri, Semi-local functions in ideal topological spaces, Journal of Advanced Research in Pure Mathematics, 2.1 (2010b), 36-42. [26] M. Khan and T. Noiri, On sgI-closed sets in ideal topological spaces, International Electronics Journal of Pure and Applied Mathematics, 3.1 (2010-2011), 29-38. · Zbl 1399.54012 · doi:10.20454/jast.2010.202 [27] M. K. R. S. V. Kumar, Between \(g^*\)-closed sets and \(g\)-closed sets, Antarctica journal mathematics 3.1 (2006), 43-65. · Zbl 1169.54313 [28] M. K. R. S. V. Kumar, On \(\check{g} \)-closed sets in topological spaces, Bull. Allah Mathematics Soc., 18 (2003), 99-112. · Zbl 1089.54502 [29] K. Kuratowski, Topology, Vol. I, Academic Press, New York, 1966. · Zbl 0158.40802 [30] N. Levine, Semi-open sets and semi-continuity in topological spaces, American Mathematical Monthly, 70 (1963), 36-41. · Zbl 0113.16304 · doi:10.1080/00029890.1963.11990039 [31] N. Levine, A decomposition of continuity in topological spaces, American Mathematical Monthly, 68 (1961), 44-46. · Zbl 0100.18601 · doi:10.2307/2311363 [32] N. Levine, Generalized closed sets in topology, Rendiconti del Circolo Matematica di Palermo Series 2, 19 (1970), 89-96. · Zbl 0231.54001 · doi:10.1007/BF02843888 [33] H. Maki, R. Devi, and K. Balachandran, Associated topologies of generalized \(\alpha \)-closed sets and \(\alpha \)-generalized closed sets, Memoirs of the Faculty of Science Kochi University Series A Mathematics, 15 (1994), 51-63. · Zbl 0821.54002 [34] A. S. Mashhour, M. E. El-Monsef, and S. N. El-Deeb, On precontinuous and weak precontinuous functions, Proc. Math. Phys. Soc. Egypt, 53 (1982), 47-53. · Zbl 0571.54011 [35] M. N. Mukherjee, R. Bishwambhar, and R. Sen, On extension of topological spaces in terms of ideals, Topology and its Appl., 154 (2007), 3167-3172. · Zbl 1132.54016 · doi:10.1016/j.topol.2007.08.014 [36] M. Murugalingam, A study of semi generalized topology, Ph.D. Thesis, Manonmaniam Sundaranar University Tirunelveli Tamil Nadu India, 2005. [37] A. A. Nasef and R. A. Mahmoud, Some applications via fuzzy ideals, Chaos, Solitons and Fractals, 13 (2002), 825-831. · Zbl 1028.54012 · doi:10.1016/S0960-0779(01)00058-3 [38] R. L. Newcomb, Topologies which are compact modulo an ideal, Ph.D. Dissertation. University of California - Santa Barbara, 1967. [39] T. Noiri, H. Maki, and J. Umehara, Generalized preclosed functions, Indian Journal of Pure and Applied Mathematics, 19 (1998), 13-20. · Zbl 0894.54014 [40] T. Noiri and V. Popa, Between \(*\)-closed and I-g-closed sets in ideal topological spaces, Rendiconti del Circolo Matematico di Palermo, 59 (2010), 251-260. · Zbl 1198.54002 [41] N. Palaniappan and K. C. Rao, Regular generalized closed sets, Kyungpook Mathematics Journal, 33 (1993), 211-219. · Zbl 0794.54002 [42] N. R. Paul, Rgi-closed sets in ideal topological spaces, International Journal of Computer Applications, 69.4 (2013), 23-27. [43] D. Ranč, In compactness modulo an ideal, Soviet Mathematics, Doklady, 13.1 (1973), 193-197. [44] O. Ravi and S. Tharmar, \(^*g\)-closed sets in ideal topological spaces, Jordan Journal of Mathematics and Statistics, 6.1 (2013), 1-13. · Zbl 1277.54006 [45] R. Vaidyanathaswamy, The localization theory in set-topology, Proceedings of the Indian Academy of Science, 20 (1945), 51-60. · Zbl 0061.39308 [46] S. Yüksel, A. Açikgöz, and T. Noiri, On \(\alpha -I\)-continuous functions, Turk. J. Math., 29 (2005), 39-51. · Zbl 1064.54026 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.