Acta Mathematica Hungarica

, Volume 140, Issue 1–2, pp 47–59 | Cite as

On generalized cluster sets of functions and multifunctions

Article

Abstract

This paper introduces and studies generalized cluster sets (g-cluster sets) of functions and multifunctions on GTS, which unifies the existing notions of cluster sets, θ-cluster sets, δ-cluster sets, S-cluster sets, s-cluster sets, p-cluster sets and many more. Several properties of the functions and multifunctions as well as their range and domain spaces are observed via degeneracies of their g-cluster sets. Characterizations of g-cluster sets through filterbases and grills on a typical class of GTS’s are also obtained. Moreover, μ-compactness of a GTS is characterized through g-cluster sets of multifunctions.

Key words and phrases

(μi,μj)-adhere (μi,μj)-converge g-cluster set g-closedness (μ1μ2,η)-continuity 

Mathematics Subject Classification

54A05 54D20 54D30 54C60 54C99 

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References

  1. [1]
    P. S. Alexandroff, On bicompact extensions of topological spaces, Mat. Sb., (N. S.), 5(47) (1939), 403–423 (in Russian). MathSciNetGoogle Scholar
  2. [2]
    P. S. Alexandroff and P. S. Urysohn, On compact topological spaces, Akad. Nauk SSSR, Trudy Mat. Inst. Steklova, 31 (1950), 1–96 (in Russian). Google Scholar
  3. [3]
    C. K. Basu and M. K. Ghosh, A new kind of cluster sets and their applications, Missouri J. Math. Sci., 21 (2009), 198–205. MathSciNetMATHGoogle Scholar
  4. [4]
    N. Bourbaki, General Topology, Chapters 1–4, Elements of Math., Springer Verlag (Berlin, 1989). CrossRefGoogle Scholar
  5. [5]
    D. A. Carnahan, Locally nearly compact spaces, Boll. Un. Mat. Ital., 6 (1972), 146–153. MathSciNetMATHGoogle Scholar
  6. [6]
    Á. Császár, Generalized topology, generalized continuity, Acta Math. Hungar., 96 (2002), 351–357. MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Á. Császár and E. Makai Jr., Further remarks on δ- and θ-modifications, Acta Math. Hungar., 123 (2009), 223–228. MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    J. Dontchev, M. Ganster and T. Noiri, On p-closed spaces, Internat. J. Math. Sci., 24 (2000), 203–212. MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    T. R. Hamlett, Cluster sets in general topology, J. London Math. Soc. (2), 12 (1976), 192–198. MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    J. E. Joseph, Multifunctions and cluster sets, Proc. Amer. Math. Soc., 74 (1979), 329–337. MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    J. C. Kelly, Bi-topological spaces, Proc. London Math. Soc., 13 (1963), 7–89. Google Scholar
  12. [12]
    A.-K. Abd El Aziz Ahmed, On Generalised Forms of Compactness, Master’s Thesis, Faculty of Science. Tanta University, Egypt, 1989. Google Scholar
  13. [13]
    N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70 (1963), 36–41. MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    G. D. Maio and T. Noiri, On s-closed spaces, Indian J. Pure and Appl. Math., 18 (1987), 226–233. MathSciNetMATHGoogle Scholar
  15. [15]
    A. S. Mashhour, M. E. Abd El-Monsef and S. N. El-Deeb, On pre-continuous and weak pre-continuous mappings, J. Math. Phys. Soc. Egypt, 53 (1982), 47–53. MathSciNetMATHGoogle Scholar
  16. [16]
    W. K. Min, Mixed weak continuity on generalized topological spaces, Acta Math. Hungar. (2011), doi: 10.1007/s10474-011-0078-2. Google Scholar
  17. [17]
    W. K. Min, A note on δ- and θ-modifications, Acta Math. Hungar. (2010), doi:  10.1007/s10474-010-0045-3. Google Scholar
  18. [18]
    M. N. Mukherjee and A. Debray, On S-cluster sets and S-closed spaces, Internat. J. Math. & Math. Sci., 23 (2000), 597–603. MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    M. N. Mukherjee and S. Raychaudhuri, A new type of cluster sets and its applications, Tamkang J. Math., 26 (1995), 327–336. MathSciNetMATHGoogle Scholar
  20. [20]
    M. N. Mukherjee and B. Roy, On p-cluster sets and their applications to p-closedness, Carpathian J. Math., 22 (2006), 99–106. MathSciNetMATHGoogle Scholar
  21. [21]
    T. Noiri, On S-closed spaces, Ann. Soc. Sci. Bruxelles Ser. I, 91 (1977), 189–194. MathSciNetMATHGoogle Scholar
  22. [22]
    T. Noiri, On S-closed spaces and S-perfect functions, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 120 (1986), 71–79. MathSciNetMATHGoogle Scholar
  23. [23]
    I. L. Reilly, On bi-topological separation properties, Nanta Math., 5 (1972), 14–25. MathSciNetMATHGoogle Scholar
  24. [24]
    M. K. Singal and A. Mathur, On nearly compact spaces, Boll. Un. Mat. Ital., 6 (1969), 63–73. MathSciNetGoogle Scholar
  25. [25]
    R. E. Smithson, Multifunctions, Nieuw. Arch. Wisk., 20 (1972), 31–53. MathSciNetMATHGoogle Scholar
  26. [26]
    T. Thompson, S-closed spaces, Proc. Amer. Math. Soc., 60 (1976), 335–338. MathSciNetGoogle Scholar
  27. [27]
    W. J. Thron, Proximity structures and grills, Math. Ann., 206 (1973), 35–62. MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    N. V. Veličko, H-closed topological spaces, Amer. Math. Soc. Transl., 78 (1968), 103–118, Mat. Sb. (N. S.), 70 (1966), 98–112. Google Scholar
  29. [29]
    J. D. Weston, Some theorems on cluster sets, J. London Math., 33 (1958), 435–441. MathSciNetMATHCrossRefGoogle Scholar
  30. [30]
    G. T. Whyburn, Analytic Topology, Amer. Math. Soc. Coll. Publs. 28 (New York, 1942). MATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2013

Authors and Affiliations

  1. 1.Department of MathematicsWest Bengal State UniversityKolkataIndia

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