Acta Mathematica Hungarica

, Volume 130, Issue 4, pp 349–357 | Cite as

Unification of generalized open sets on topological spaces

Article

Abstract

A new kind of sets called generalized μ-closed (briefly gμ-closed) sets are introduced and studied in a topological space by using the concept of generalized open sets introduced by Á. Császár. The class of all gμ-closed sets is strictly larger than the class of all μ-closed sets. Furthermore, g-closed sets (in the sense of N. Levine [17]) is a special type of gμ-closed sets in a topological space. Some of their properties are investigated. Finally, some characterizations of μg-regular and μg-normal spaces have been given.

Key words and phrases

μ-open set gμ-closed set μg-regular μg-normal space 

2000 Mathematics Subject Classification

54D10 54D15 54C08 54C10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. E. Abd El-Monsef, S. N. El-Deeb and R. A. Mahmoud, β-open sets and β-continuous mappings, Bull. Fac. Sci. Assiut Univ., 12 (1983), 77–90. MathSciNetGoogle Scholar
  2. [2]
    M. E. Abd El-Monsef, A. N. Geaisa and R. A. Mahmoud, β-regular spaces, Proc. Math. Phys. Soc. Egypt, 60 (1985), 47–52. MATHGoogle Scholar
  3. [3]
    A. Al-Omari and M. Salmi Md. Noorani, On generalized b-closed sets, Bull. Malays. Math. Sci. Soc., 32 (2009), 19–30. MATHMathSciNetGoogle Scholar
  4. [4]
    D. Andrijević, On b-open sets, Mat. Vesnik, 48 (1996), 59–64. MATHMathSciNetGoogle Scholar
  5. [5]
    S. P. Arya and T. Nour, Characterizations of s-normal spaces, Indian J. Appl. Math., 21 (1990), 717–719. MATHMathSciNetGoogle Scholar
  6. [6]
    Á. Császár, Generalized topology, generalized continuity, Acta Math. Hungar., 96 (2002), 351–357. MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Á. Császár, Generalized open sets in generalized topologies, Acta Math. Hungar., 106 (2005), 53–66. MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Á. Császár, δ- and θ-modifications of generalized topologies, Acta Math. Hungar., 120 (2008), 275–279. MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    J. Dontchev and M. Ganster, On δ-generalized closed sets and T 3/4-spaces, Mem. Fac. Sci. Kochi Univ. Ser. A, Math., 17 (1996), 15–31. MATHMathSciNetGoogle Scholar
  10. [10]
    J. Dontchev, On generalizing semi-preopen sets, Mem. Fac. Sci. Kochi Univ. Ser. A, Math., 16 (1995), 35–48. MATHMathSciNetGoogle Scholar
  11. [11]
    J. Dontchev and H. Maki, On θ-generalized closed sets, Internat. J. Math. Math. Sci., 22 (1999), 239–249. MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    E. Ekici, On γ-normal spaces, Bull. Math. Soc. Sci. Math. Roumanie, 50(98) (2007), 259–272. MathSciNetGoogle Scholar
  13. [13]
    E. Ekici and T. Noiri, On a generalization of normal, almost normal and mildly normal spaces-I, Math. Moravica, 10 (2006), 9–20. MATHMathSciNetGoogle Scholar
  14. [14]
    S. N. El-Deeb, I. A. Hasanein, A. S. Mashhour and T. Noiri, On p-regular spaces, Bull. Math. Soc. Sci. Math. R.S. Roumanie (N.S.), 27(75) (1983), 311–315. MathSciNetGoogle Scholar
  15. [15]
    M. Ganster and M. Steiner, On b τ-closed sets, Appl. Gen. Topol., 8 (2007), 243–247. MATHMathSciNetGoogle Scholar
  16. [16]
    J. L. Kelley, General Topology, Van Nostrand (New York, 1955). MATHGoogle Scholar
  17. [17]
    N. Levine, Generalized closed sets in topology, Rend. Circ. Mat. Palermo, (2), 19 (1970), 89–96. MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70 (1963), 36–41. MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    S. N. Maheshwari and R. Prasad, On s-normal spaces, Bull. Math. Soc. Math. R.S. Roumanie, 22(70) (1978), 27–29. MathSciNetGoogle Scholar
  20. [20]
    S. N. Maheshwari and R. Prasad, On s-regular spaces, Glasnik Mat. Ser. III, 10(30) (1975), 347–350. MathSciNetGoogle Scholar
  21. [21]
    R. A. Mahmoud and M. E. Abd El-Monsef, β-irresolute and β-topological invariants, Proc. Pakistan Acad. Sci., 27 (1990), 285–296. MathSciNetGoogle Scholar
  22. [22]
    G. Di Maio and T. Noiri, On s-closed spaces, Indian J. Pure Appl. Math., 18 (1987), 225–233. Google Scholar
  23. [23]
    H. Maki, R. Devi and K. Balachandran, Associated topologies of generalized α-closed sets and α generalized closed sets, Mem. Fac. Sci. Kochi Univ. (Math.), 15 (1994), 51–63. MATHMathSciNetGoogle Scholar
  24. [24]
    A. S. Mashhour, M. E. Abd El-Monsef and S. N. El-Deeb, On precontinuous and weak precontinuous mappings, Proc. Math. Phys. Soc. Egypt, 53 (1982), 47–53. MATHMathSciNetGoogle Scholar
  25. [25]
    T. M. Nour, Contributions to the Theory of Bitopological Spaces, Ph.D. Thesis (Delhi, 1989). Google Scholar
  26. [26]
    G. Navalagi and Md. Hanif, On θ gs-neighbourhoods, Indian J. Math. Math. Sci., 4 (2008), 21–31. MathSciNetGoogle Scholar
  27. [27]
    O. Njåstad, On some classes of nearly open sets, Pacific J. Math., 15 (1965), 961–970. MATHMathSciNetGoogle Scholar
  28. [28]
    J. H. Park, B. Y. Lee and M. J. Son, On δ-semiopen sets in topological space, J. Indian Acad. Math., 19 (1997), 59–67. MATHMathSciNetGoogle Scholar
  29. [29]
    J. H. Park, Y. B. Park and B. Y. Lee, On gp-closed sets and pre gp-continuous functions, Indian J. Pure Appl. Math., 33 (2002), 3–12. MATHMathSciNetGoogle Scholar
  30. [30]
    J. H. Park, Strongly θ-b-continuous functions, Acta Math. Hungar., 110 (2006), 347–359. MATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    J. H. Park, D. S. Song and R. Saadati, On generalized δ-semiclosed sets in topological spaces, Chaos Solitons and Fractals, 33 (2007), 1329–1338. MATHCrossRefMathSciNetGoogle Scholar
  32. [32]
    S. Raychaudhuri and M. N. Mukherjee, On δ-almost continuity and δ-preopen sets, Bull. Inst. Math. Acad. Sinica, 21 (1993), 357–366. MATHMathSciNetGoogle Scholar
  33. [33]
    N. V. Veličko, H-closed topological spaces, Mat. Sb., 70 (1966), 98–112. MathSciNetGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2010

Authors and Affiliations

  1. 1.Yatsushiro-shiJapan
  2. 2.Department of MathematicsWomen’s Christian CollegeKolkata-India

Personalised recommendations