A Unified Theory for Certain Weak Forms of Open Sets and Their Variant Forms

  • Roy, Bishwambhar ;
  • Seny, Ritu
  • Received : 2011.03.07
  • Accepted : 2012.06.04
  • Published : 2012.12.23


The purpose of the present paper is towards working out a unified version of the study of certain weak forms of generalized open sets and their neighbouring forms, as are already available in the literature. In terms of an operation, as initiated by $\acute{A}$. Cs$\acute{a}$sz$\acute{a}$r, we introduce unified definitions of ${\wedge}_{\psi}$-sets, ${\vee}_{\psi}$-sets, $g{\cdot}{\wedge}_{\psi}$-sets and $g{\cdot}{\vee}_{\psi}$-sets and derive results concerning them.


Operation;${\psi}$-closure;${\psi}$-open set;${\wedge}_{\psi}$-sets;$g{\cdot}{\wedge}_{\psi}$-sets


  1. M. E. Abd El-Monsef, S. N. El-Deeb and R. A. Mahmoud, ${\wedge}_{s}$-open sets and ${\vee}_{s}$-continuous mappings, Bull. Fac. Sci. Assiut Univ., 12(1983), 77-90.
  2. P. Alexandroff, Diskrete Raume, Math. Sb., 2(1937), 501-508.
  3. M. Caldas and J. Dontchev, G:${\Lambda}_{s}$-sets and G: ${\nu}_{s}$-sets, Mem. Fac. Sci. Kochi Univ. (Math.), 21(2000), 21-30.
  4. M. Caldas, D. N. Georgiou and S. Jafari, Study of $({\wedge},{\alpha})$-closed sets and related notions in topological spaces, Bull. Malays. Math. Sci. Soc., 30(2)(1)(2007), 23-36.
  5. M. Caldas, S. Jafari and T. Noiri, On ${\wedge}_{b}$ sets and the associated topology ${\tau}^{{\wedge}_{b}}$, Acta Math. Hungar., 110(2006), 337-345.
  6. A. Csaszar, Generalized open sets, Acta. Math. Hungar., 75(1-2)(1997), 65-87.
  7. J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.
  8. E. Ekici and B. Roy, New generalized topologies on generalized topological spaces due to Csaszar, Acta Math. Hungar., 132(2011), 117-124.
  9. M. Ganster, S. Jafari and T. Noiri, On pre-${\wedge}$-sets and pre-${\wedge}$-sets, Acta Math. Hungar., 95(2002), 337-343.
  10. S. Jafari, Rare ${\alpha}$-continuity, Bull. Malays. Math. Sci. Soc., 28(2)(2005), 157-161.
  11. M. Kucuk and I. Zorlutuna, A unified theory for weak separation properties, Internat. J. Math and Math. Sci., 24(11)(2000), 765-772.
  12. M. Kucuk and I. Zorlutuna, A unification on compactness and closedness, Soochow J. Math., 29(3)(2003), 221-233.
  13. N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70(1963), 36-41.
  14. R. A. Mahmoud and M. E. Abd El-Monsef, ${\beta}$-irresolute and ${\beta}$-topological invariant, Proc. Pakistan Acad. Sci., 27(3)(1990), 285-296.
  15. H. Maki, Generalized ${\wedge}$-sets and the associated closure operator, Special issue in commemoration of Prof. Kazusada IKEDA's retirement, (1986), 139-146.
  16. 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.
  17. T. Noiri and E. Hatir, ${\Lambda}$sp-sets and some weak separation axioms, Acta. Math. Hungar., 103(3)(2004), 225-232.
  18. S. Raychaudhuri and M. N. Mukherjee, On -almost continuity and ${\delta}$-preopen sets, Bull. Inst. Math. Acad. Sinica, 21(1993), 357-366.
  19. B. Roy and M. N. Mukherjee, A unified theory for $R_{0},R_{1}$ and certain other separation properties and their variant forms, Bol. Soc. Paran. Mat., 28(2)(2010), 9-18.
  20. I. L. Reilly and M. K. Vamanamurthy, On semi compact spaces, Bull. Malays. Math. Soc., 7(2)(1984), 61-67.