Have a personal or library account? Click to login
A Unified Treatment of Generalized Closed Sets in Topological Spaces Cover

A Unified Treatment of Generalized Closed Sets in Topological Spaces

By: Emilia Przemska  
Open Access
|Nov 2023

References

  1. ABD EL-MONSEF, E.—EL-DEEB, N.—MAHMOUD, R. A.: β-open sets and β-continuous mapping, Bull. Fac. Sci. Assiut Univ. 12 (1983) 77–90.
  2. ABD EL-MONSEF, M. E.—MAHMOUD, R. A.—LASHIN, E. R.: β-closure and β-interior, J. Fac. Ed. Ain Shams Univ. 10 (1986) 235–245.
  3. ABD EL-MONSEF, M. E.—EL-DEEB, S. N.—MAHMOUD, R. A.: β-open sets and β-continuous mappings, Bull. Fac. Sci. Assiut Univ. 12 (1983), 77–90.
  4. AL-OMARI, A.—NOORANI, M.—MD, S.: On generalized b-closed sets, Bull. Malaysian Math.Sci.Soc. 32 (2009), no. 1, 19–30.
  5. AL-OMARI, A.—NOIRI, T.: A unified theory of generalized closed sets in weak structures, Acta Math. Hungar. 135 (2012), 174–183.
  6. AMEEN,Z.A.: On types of generalized closed sets, Journal of Taibah University for Science 12, no. 3 (2018), 290–293.
  7. ANDRIJEVIC, D.: Semi-preopen sets,Mat.Vesnik 38 (1986), no. 93, 24–32.
  8. ANDRIJEVIC, D.: On b-open sets,Mat. Vesnik 48 (1996), no. 1–2, 59–64.
  9. ARYA, S. P.—NOUR, T. M.: Characterizations of s-normal spaces, Indian J. Pure Appl. Math. 21 (1990), no. 8, 717–719.
  10. AZZAM, A. A.: A new closed set in topological spaces, Math. Probl. Eng. 2021,Article ID 6617224, 4 pp.
  11. BENCHALLI, S. S.: Generalized ωα-closed sets in topological spaces,J.New Results in Science, 3 (2014), no. 7, 7–19.
  12. BENCHALLI, S. S.—WALI, R. S.: On RW-closed sets in topological spaces, Bull. Malaysian Math. Sci. Soc. 30 (2007), no. 2, 99–110.
  13. BHATTACHARYYA, P.: Semi-generalized closed sets in topology, Indian J. Math. 29 (1987), no. 3, 375–382.
  14. BHATTACHARYYA, P.: On generalized regular closed sets,Int.J.Contemp.Math. Sciences, 6 (2011), no. 3, 145–152.
  15. CAMERON, D. E.: Properties of S-closed spaces, Proc. Amer. Math. Soc. 72 (1978), no. 3, 581–586.
  16. CAO, J.—GANSTER, M.—REILLY, I.: Submaximality, extremal disconnectedness and generalized closed sets, Houston J. Math. 24 (1998), no. 4, 681–688.
  17. CAO, J.—GANSTER, M.—REILLY, I.: On generalized closed sets, Topology Appl. 123 (2002), no. 1, 37–46.
  18. CAO, J.—GREENWOOD, S.—REILLY, I.: Generalized closed sets: a unified approach, Appl.Gen.Topol. 2 (2001), no. 2, 179–189.
  19. CORSON, H. H.—MICHAEL, E.: Metrizability of certain countable unions, Illinois J. Math. 8 (1964), 351–360.
  20. CROSSLEY, S. G.—HILDEBRAND, S. K.: Semi-closure, Texas J. Sci. 22 (1971), 99–112.
  21. CSÁSZÁR,Á.: Weak structures, Acta Math. Hungar. 131, (2011).
  22. CSÁSZÁR,Á.: Generalized topology, generized continuity, Acta Math. Hungar. 96 (2002), no. 4, 351–357.
  23. DEVI, B.M. —PADMAPRIYA, G.: Regularĝ-closed sets in topological spaces, J. Appl. Sci. Comput. 5 (2018), no. 9, 896–900.
  24. DEVI, B. M.—PADMAPRIYA, G.: Regularĝ-closed sets in topological spaces, J. Appl. Sci. Comput. 5 (2018), no. 9, 896–900.
  25. DEVI, R.—SELVAKUMAR, A.—JAFARI, S.: ̃Gα-closed sets in topological spaces,Asia Mathe. 3 (2019), no. 3, 16–22.
  26. DI, G.—NOIRI, T.: On S-closed spaces, Indian J. Pure Appl. Math., 18 (1987), no. 3, 226–233.
  27. DONTCHEV, J.: On generalizing semi-preopen sets,Mem. Fac. Sci.Kochi Univ.Ser. A (Math.) 16 (1995), 35–48.
  28. DONTCHEV, J.: On some separation axioms associated with the α-topology,Mem. Fac. Sci. Kochi Univ. Ser. A (Math.) 18 (1997), 31–35.
  29. DONTCHEV, J.: Survey on preopen sets, Meetings on Topological Spaces, Theory and Applications, Yatsushiro College of Technology 18, preprint math/9810177, 1998.
  30. EL-ATIK, A. A.: A Study of Some Types of Mappings on Topological Spaces.Master’s Thesis, Faculty of Science, Tanta University, Tanta, Egypt, 1997.
  31. EL-DEEB, S. N.—HASANEIN, I. A.—MASHHOUR, A. S.—NOIRI, T.: On p-regular spaces, Bull. Math. Soc. Sci. Math. R. S. Roumanie 27 (1983), no. 75, 311–315.
  32. EL-MAGHRABI, A. I.: More on γ-generalized closed sets in topology, J. Taibah University for science 7 (2013), no. 3, 114–119.
  33. EL-MAGHRABI, A. I.—NASEF, A. A.: Between semi-closed and GS-closed sets, J. Taibah University for Science 2 (2009), no. 1, 78–87.
  34. ERDAL, E.: On γ-normal spaces, Bull. Math. Soc. Sci. Math. Roumanie 50(98) (2007), no. 3, 259–272.
  35. FUKUTAKE, T.—NASEF, A. A.—EL-MAGHRABI, A. I.: Some topological concepts via γgeneralized closed sets, Bull. Fukuoka Univ. Educt. Part III 52 (2003), 1–9.
  36. GANSTER, M.—STEINER, M.: On bT-closed sets, Appl. Gen. Topol. 8 (2007), no. 2, 243–247.
  37. GARGOURI, R.—REZGUI, A.: A unification of weakening of open and closed subsets in a topological space, Bull. Malays. Math. Sci. Soc. 40 (2017), 1219–1230.
  38. GNANAMBAL, Y.: On generalized preregular closed sets in topological spaces, Indian J. Pure Appl. Math. 28 (1997), no. 3, 351–360.
  39. HAMMED, D. M.: On arps–closed sets in topological spaces, Engineering and Technology Journal Part (B) Scientific, 32 (2014), no. 2, 271–286.
  40. HELEN, P. M.—KULANDHAI THERESE, M.: A (gsp)∗-closed sets in Topological spaces,IJMTT 6 (2014), 75–78.
  41. HOLLIYAVAR, M. M.—RAYANAGOUDAR, T. D.—SARIKA, M. P.: On Semi pre generalized ωα-closed sets in topological spaces, Glob. J. Pure and Appl. Math. 13 (2017), no. 10, 7627–7635.
  42. INDRANI, K.—SHATISHMOHAN, P.—RAJENDRAN, V.: On gr∗-closed sets in topological spaces, Int. J. Math. Trends and Technology (2014), 142–148.
  43. ITTANAGI, B. M.—GOVARDHANA REDDY, H. G.: On gg-Open Sets in topological sce, J. Comput. Math. Sci. 8 (2017), no. 9, 413–423.
  44. IYAPPAN, D.—NAGAVENI, N.: On semi generalized b-closed set,Nat.Sem.Mat.& Comp. Sci, (2010), Proc. 6.
  45. JAFARI, S.: Pre g∗-closed sets in topological spaces, J. Adv. Stud. Topology 3 (2012), no. 3, 55–59.
  46. JANAKI, C.—THOMAS, R.: On R∗-Closed sets in topological spaces, Int. J. Math. Archive 3 (2012), no. 8, 3067–3074.
  47. JANKOVIČ, D.: On semi separation properties, Indian J. Pure Apll. Math. 16 (1985), 957–964.
  48. JAYAKUMAR, P.—MARIAPPA, K.—SEKAR, S.: On generalized gp∗ closed set in topological spaces, Int.J.Math. Anal. 33 (2013), no. 7, 75–86.
  49. JOHN, M. S.: On ω-closed sets in topology, Acta Ciencia Indica 4 (2000), 389–392.
  50. KALIMSKY, E. D.—KOPPERMAN, R.—MEYER, P. R.: Computer graphics and connected topologies on finite ordered sets, Topology and its Applications 36 (1990), 1–17.
  51. KANNAN, K.—NAGAVENI, N.: On ˆ β-generalized closed sets and open sets in topological spaces, Int.J.Math. Anal. 6 (2012), no. 57, 2819–2828.
  52. KANNAN, K.RAJALAKSHMI, D.—SHATHYAASHREE, C. K.: Strongly s∗g∗-closed sets,Int.J.Pure Appl.Math. 102 (2015), no. 4, 643–652.
  53. KONY, T. Y.—KOPPERMAN, R.—MEYER, P. R.: A topological approach to digital toplogy, Amer. Math. Monthly 98 (1991), 901–917.
  54. KUMAR, M. V.:ĝ-closed sets in topological spaces, Bull. Allahabad Math. Soc. 18 (2003), 99–112.
  55. KUMAR, M. V.: g#-semi-closed sets in topological spaces, Indian J. Math. 44 (2002), 73–87.
  56. KUMAR, M. V.: Between semi-closed sets and semi-pre-closed set, Rend. Istit. Mat. Univ. Trieste 32 (2000), 25–41.
  57. KUMAR, M. V.: Between closed sets and g-closed sets,Mem.Fac.Sci.Kochi Univ. Ser. A Math. 21 (2000), 1–19.
  58. KUMAR, M. V.:, g#-closed sets in topological spaces, Mem.Fac. Sci.Kochi Univ.Ser. AMath. 24 (2003), 1–13.
  59. KUMAR, M. V.:, #g-semi-closed sets in topological spaces, Antarctica J. Math. 2 (2005), no. 2, 201–222.
  60. LEELAVATHI, S. T.—MARIASINGAM, M.: On #α-regular generalized closed in topological spaces,IJMTT 56 (2018), no. 8, 551–557.
  61. LEVINE, N.: Generalized closed sets in topology, Rend. Circ. Mat. Palermo (2) 19 (1970), no. 1, 89–96.
  62. LEVINE, N,: Semi-open sets and semi-continuity in topological spaces,Amer. Math. Monthly 70 (1963), no. 1, 36–41.
  63. MAKI, H.: Every topological space is pre-T1/2,Mem.Fac.Sci.Kochi Univ. Ser. A Math. 17 (1996), 33–42.
  64. MAKI, H.: Associated topologies of generalized-closed sets and generalized closed sets, Mem. Fac. Sci. Kochi Univ. Ser. A. Math. 15 (1994), 51–63.
  65. MAKI, H.—DEVI, R.—BALACHANDRAN, K.: Generalized α-closed sets in topology, Bull. Fukuoka Univ. Ed. Part III 42 (1993), 13–21.
  66. MARIAPPA, K.—SEKAR, S.: On regular generalized b-closed set,Internat. J. Math. Anal. 7 (2013), no. 13, 613–624.
  67. MARY, T. S. I—THANGAVELU, P.: On regular pre-semi closed sets in topological spaces, J. Math. Sci. & Comput. Appl. 1 (2010), no. 1, 9–17.
  68. MASHHOUR, A. S.—HASANEIN, I. A.—EL-DEEB, S. N.: α-continuous and α-open mappings, Acta Math. Hungar. 41 (1983), 213–218.
  69. MASHHOUR, A. S.: On precontinuous and weak precontinuous mappings,Proc. Math. Phys.Soc.Egypt. 53 (1982), 47–53.
  70. MEENA, K.: On pre generalized star b-closed set in topological spaces,Iaetsd J. Adv. Res. Appl. Sci. 5 (2018), no. 4, 322–327.
  71. MISHRA, S.—JOSHI, V.—BHARDWAJ, N.: On generalized pre regular weakly (gprw)--closed sets in topological spaces, Int. Math. Forum 7 (2012), no. 37–40, 1981–1992.
  72. MISHRA, S.—BHARDWAJ, N.—JOSHI, V.: On regular generalized weakly (rgw)-closed sets in topological spaces, Int.J.Math. Anal. 6 (2012), no. 39, 1939–1952.
  73. MOHAN, V.—BASAVARAJ, M. I.: On semi α-regular weakly closed set in topological spaces,Int. J. of Math. Archive 8 (2017), no. 7, 197–204.
  74. MUKUNDHAN, C.—NAGAVENI, N.: A weaker form of a generalized closed set, Int. J. Contemp. Math. Sciences, 6 (2011), no. 20, 949–961.
  75. NAGAVENI, N.: Studies on Generalizations of Homeomorphisms in Topological Spaces. Diss. Ph. D. Thesis, Bharathiar University Coimbatore, 1999.
  76. NAGAVENI, N.—NARMADHA, A.: On regular b-closed sets in topological spaces, In: Heber International Conference on Applications of Mathematics and Statistics 2012, pp. 5–7.
  77. NARMADHA, A.—NAGAVENI, N.— NOIRI, T.: On regular b-open sets in topological spaces, Int. J. Math, Anal. 7 (2013), no. 19, 937–948.
  78. NASEF, A.—MAREAY, R.: New topological approach of generalized closed sets,J.New Results in Sci. 4 (2015), no. 9, 67–78.
  79. NARASIMHAN, D.—SUBHAA, R.: gs-closed set, Int. J. Pure Appl. Math. 119 (2018), no. 6, 209–218.
  80. NARASIMHAN, D.—JAYANTHI, J.: On gp-closed set in topological spaces, Int.J.Pure Appl. Math. 119 (2018), no. 6, 199–208.
  81. EL NASCHIE, M. S.: On the uncertainty of Cantorian geometry and two, slit experiment, Chaos, Solitons & Fractals 9 (1998), no. 3, 517–529.
  82. NAVALAGI, G.: Properties of G∗-closed sets in topological spaces, Int. J. Recent Sci. Research 9 (2018), no. 8, 28176–28180.
  83. NAVALAGI, G. B.—TIPPESHI.MARIGOUDAR, V.: On pre generalized ωα-closed sets in topological spaces, Int. J. Math. Trends and Technology (IJMTT) 58 (2018), no. 2, 94–97.
  84. NAVALAGI, G.—BHAVIKATTI, K. M.: Beta weakly generalized cosed sets in topology, J. Comput. Math. Sci. 9 (2018), no. 5, 435–446.
  85. NAVALAGI, G.—MEGALAMANI, S. B.: g∗γ-closed sets in topological spaces,Int. J. Management & Social Science 6 (2018), no. 6, 151–160.
  86. NAVALAGI, G.—CHARANTIMATH, R. G.: Properties of wgr-closed sets in topological spaces, Int. J. Engnr. Res. Development 14 (2018), no. 7, 58–62.
  87. NAVALAGI, G.— MARIGOUDAR, T. V.: On pre generalized ωα-closed sets in topological spaces, Glob. J. Pure Appl. Math. 13 (2017), no. 9, 5491–5503.
  88. NJ ˙aSTAD, O.: On some classes of nearly open sets,Pacific J.Math. 15 (1965), no. 3, 961–970.
  89. NOIRI, T.—POPA, V.: A note on modifications of rg-closed sets in topological spaces, Cubo (Temuco) 15 (2013), no. 2, 65–70.
  90. NOIRI, T.—ROY, B.: Unification of generalized open sets on topological spaces,Acta Math. Hungar. 130 (2011), 349–357
  91. ORE, O.: Galois connexions, Trans.Amer. Math.Soc. 55 (1944), 493–513.
  92. PALANIAPPAN, N.: Regular generalized closed sets, Kyungpook Math. J. 33 (1993), no. 2, 211–219.
  93. PARIMALA, M.: On αω-closed sets in topological spaces, Int. J. Pure Appl. Math. 115 (2017), no. 5, 1049–1056.
  94. PARK, J. K.—PARK, J. H.: Mildly generalized closed sets, almost normal and mildly normal spaces, Chaos, Solitons & Fractals 20 (2004), no. 5, 1103–1111.
  95. PARK,J.H.—PARK, Y.B.—LEE, B.Y.: On gp-closed sets and pre gp-continuous functions, Indian J. Pure Appl. Math. 33 (2002), 3–12.
  96. POPA, V.: On M-continuous functions, Ann. Univ. Dunarea de Jos Galati Fasc. II Mat. Fiz. Mec. Teor. 18 (2000), no. 23, 31–41.
  97. PUSHPALATHA, A.—ANITHA, K.: g∗s-closed sets in topological spaces,Internat. J. Contemp. Math. Sci. 6 (2011), no. 19, 917–929.
  98. PATIL, P. H.—PATIL, P. G.: Generalized pre α-closed sets in topology, Journal of New Theory 20, (2018), 48–56.
  99. PRZEMSKA, E.: The lattices of families of regular sets in topological spaces, Math. Slovaca 70 (2020), no. 2, 477–488.
  100. RAJENDIRAN, R.—THAMILSELVAN, M.: g∗s∗ closed sets in topological spaces, Int. J. Math. Anal. 8 (2014), no. 39, 1919–1930.
  101. RAJESHWARI, K.—RAYANAGOUDAR, T. D.—PATIL, S. M.: On semi generalized ωα-closed sets in topological spaces, Global J. Pure Appl. Math. 13 (2017), no. 9, 5491–5503.
  102. RAJESHWARI, K.—RAYANAGOUDAR, T. D.: On αg∗-preclosed sets in topological spaces, Int. J. Engnr. Sci. Math. 5 (2016), no. 4, 1–8.
  103. RASSIAS, TH. M.—SZÁZ,Á.: Ordinary,super andhyper relators can beusedtotreat the various generalized open sets in a unified way, In: (N.J. Daras and Th.M. Rassias, Eds.), Approximation and Computation in Science and Engineering, Springer Optimization and Its Applications, Vol. 180, Springer Nature Switzerland AG, (2022), pp. 709–782.
  104. RAWI, O.—GANSAN, S.: g-closed sets in topology, Internat. J. Comput. Sci. & Engnr. Technology (IJCSET) 2 (2011), no. 3, 330–337.
  105. RAVI, O.—RAJASEKARAN, I.—MURUGESAN, S.: On β-normal spaces,Int. J.Math. Appl. 3 (2015), no. 2, 35–44.
  106. ROSAS, E.—GNANAMBAL, I.: ANoteon αgrw-closed Sets, Eur. J. Pure Appl. Math.9 (2016), no. 1, 27–33.
  107. SARANYA, S.—BAGEERATHI, K.: Semi generalized closed sets in Topological Spaces, Int. J. Math. Trends and Technology, 36 (2016), no. 3, 23–27.
  108. SARAVANAKUMAR, D.—SATHISHKUMAR, K. M.: On a class of αg∗∗-closed sets in topological spaces and some mappings, Int.J.Sci.Res.Publ. 2 (2012), no. 6, 1–5.
  109. SARSAK, M. S.—RAJESH, N.: π-generalized semi-preclosed sets,Int. Math. Forum. 5 (2010), 573–578.
  110. SEKAR, S.—JOTHILAKSHMI, B.: On semi generalized starb-closed sets in topological spaces, Int. J. Pure and Appl. Math. 111 (2016), no. 3, 419–428.
  111. SEKAR, S.—KUMAR, G.: On gαr closed set in topological spaces,Int.J. Pure and Appl. Math. 108 (2016), no. 4, 791–800.
  112. SEKAR, S.—KUMAR, G.: On gαr closed set in topological spaces,Int.J. Pure and Appl. Math. 108 (2016), no. 4, 791–800.
  113. SEKAR, S.—LOGANAYAGI, S.: On generalized b star-closed set in topological spaces, Malaya J. Math. 5 (2017), no. 2, 401–406.
  114. SEKAR, C.—RAJAKUMARI, J.: A new notion of generalized closed sets in topological spaces, Int. J. Math. Trends and Technology 36 (2016), no. 2, 124–129.
  115. SEKAR, S.—BRINDHA, R.: On pre generalized b-closed set in topological spaces, Int. J. Pure and Appl. Math. 111 (2016), no. 4, 577–586.
  116. SELVANAYAKI, N.—GNANAMBAL, I.: On α-generalized regular weakly closed sets in topological spaces, Department of Mathematics Northwest University 9 (2013), no. 1, 49–55.
  117. SREEJAA, D.—SASIKALAB, S.: Generalized star semi regular closed sets in topological spaces, Malaya J. Mat. (MJM) 1 (2015), 42–56.
  118. STONE, M. H.: Applications of the theory of Boolean rings to general topology,Trans. Amer. Math. Soc. 41 (1937), no. 3, 375–481.
  119. SUNDARAM, P.: ˜g-semi-closed sets in topological saces, Math. Pannon. 18 (2007), no. 1, 51–61.
  120. TARSKI, A.: Fundamentale begriffe der methodologie der deduktiven wissenschaften, Monats. Math. Phys. 37 (1930), no. 1, 361–404.
  121. THIVAGAR, M. L.—PAUL, N. R.—JAFARI, S.: On new class of generalized closed sets, Ann. University of Craiova-Math. Comput. Sci. Ser. 38 (2011), no. 3, 84–93.
  122. VADIVEL, A.—VAIRAMANICKAM, K.: rgα-closed sets and rgα-open sets in topological spaces, Int.J.Math. Anal. 3 (2009), no. 37, 1803–1819.
  123. VINAYAGAMOORTHI, L.—NAGAVENI, N.: On generalized-αbclosed sets, Proceeding ICMD-Allahabad Puspha Publication 1 (2010), 2010–2011.
  124. WAD-DIRASAT, M. L. B.: Generalized b-closed sets, Department of Mathematics & Statistics Mu’tauh University 5 (2008), no. 1, 27–39.
  125. WALI, R. S.—BAJIRAO, P. K.: On (RW )* closed sets in topological spaces,J.Comput. Math. Sci. 7 (2016), no. 4, 192–202.
  126. WALI, R. S.—PRABHAVATI, S. M.: On α regular ω-open sets in topological spaces, J. Comput. Math. Sci. 5 (2014), no. 6, 490–499.
  127. WALI, R. S.—DEMBRE, V.: On pre generalızed pre regular weakly closed sets in topologıcal spaces, J. Comput. Math. Sci. 6 (2015), no. 2, 113–125.
  128. WALI, R. S.—DEMBRE, V.: On pre generalized pre regular weakly open sets and pre generalized pre regular weakly neighbourhoods in topological spaces, Ann. Pure App. Math. 10 (2015), no. 1, 15–20.
DOI: https://doi.org/10.2478/tmmp-2023-0028 | Journal eISSN: 1338-9750 | Journal ISSN: 12103195
Language: English
Page range: 121 - 154
Submitted on: Jan 18, 2023
Published on: Nov 16, 2023
Published by: Slovak Academy of Sciences, Mathematical Institute
In partnership with: Paradigm Publishing Services
Publication frequency: 3 issues per year

© 2023 Emilia Przemska, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.