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Ptr,s)-CLOSED SPACES AND PRE-(ωr,s)t-θf-CLUSTER SETS

  • Received : 2010.01.11
  • Published : 2011.01.31

Abstract

Using (r, s)-preopen sets [14] and pre-${\omega}_t$-closures [6], a new kind of covering property $P^t_{({\omega}_r,s)}$-closedness is introduced in a bitopological space and several characterizations via filter bases, nets and grills [30] along with various properties of such concept are investigated. Two new types of cluster sets, namely pre-(${\omega}_r$, s)t-${\theta}_f$-cluster sets and (r, s)t-${\theta}_f$-precluster sets of functions and multifunctions between two bitopological spaces are introduced. Several properties of pre-(${\omega}_r$, s)t-${\theta}_f$-cluster sets are investigated and using the degeneracy of such cluster sets, some new characterizations of some separation axioms in topological spaces or in bitopological spaces are obtained. A sufficient condition for $P^t_{({\omega}_r,s)}$-closedness has also been established in terms of pre-(${\omega}_r$, s)t-${\theta}_f$-cluster sets.

References

  1. A. A. K. Abd El-Aziz, On generalized forms of compactness, Master's thesis, Faculty of Science, Taunta University, Egypt, 1989.
  2. M. E. Abd El-Monsef, S. N. El-Deeb, and R. A. Mahmoud, ${\beta}-open$ sets and ${\beta}-continuous$ mapping, Bull. Fac. Sci. Assiut Univ. A 12 (1983), no. 1, 77-90.
  3. K. Al-Zoubi and B. Al-Nashef, The topology of ${\omega}-open$ subsets, Al-Manarah Journal 9 (2003), no. 2, 169-179.
  4. C. K. Basu and M. K. Cohosh, ${\beta}-closed$ spaces and ${\beta}-{\theta}-subclosed$ graphs, Eur. J. Pure Appl. Math. 1 (2008), no. 3, 40-50.
  5. C. K. Basu and M. K. Cohosh, Locally ${\beta}-closed$ spaces, Eur. J. Pure Appl. Math. 2 (2009), no. 1, 85-96.
  6. C. K. Basu and B. M. Uzzal Afsan, $P_{\omega}-Closedness$ and Its generalization with respect to a grill, communicated.
  7. C. K. Basu, B. M. Uzzal Afsan, and S. S. Mandal, Degeneracy of some cluster sets, Italian J. of Pure and Appl. Math. (accepted and to appear).
  8. C. K. Basu, B. M. Uzzal Afsan, and S. S. Mandal, Functions with strongly ${\beta}-{\theta}-closed$ graphs, Mathematica (Cluj), Tome 51(74), No. 2 (Dec 2009), 119-128.
  9. S. Bose and D. Sinha, Almost open, almost closed, ${\theta}-continuous$ and almost quasi-compact mappings in bitopological spaces, Bull. Calcutta Math. Soc. 73 (1981), no. 6, 345-354.
  10. J. Dontchev, M. Ganster, and T. Noiri, On p-closed spaces, Int. J. Math. Math. Sci. 24 (2000), no. 3, 203-212. https://doi.org/10.1155/S016117120000226X
  11. H. Z. Hdeib, ${\omega}-closed$ mappings, Rev. Colombiana Mat. 16 (1982), no. 1-2, 65-78.
  12. H. Z. Hdeib, ${\omega}-continuous$ functions, Dirasat Journal 16 (1989), no. 2, 136-153.
  13. D. Jankovic, I. Reilly, and M. Vamanamurthy, On strongly compact topological spaces, Questions Answers Gen. Topology 6 (1988), no. 1, 29-40.
  14. M. Jelic, A decomposition of pairwise continuity, J. Inst. Math. Comput. Sci. Math. Ser. 3 (1990), no. 1, 25-29.
  15. J. E. Joseph, Multifunctions and cluster sets, Proc. Amer. Math. Soc. 74 (1979), no. 2, 329-337. https://doi.org/10.1090/S0002-9939-1979-0524312-5
  16. C. G. Kariofillis, On minimal pairwise Hausdorff bitopological spaces, Indian J. Pure Appl. Math. 19 (1988), no. 8, 751-760.
  17. J. C. Kelly, Bitopological spaces, Proc. London Math. Soc. (3) 13 (1963), 71-89. https://doi.org/10.1112/plms/s3-13.1.71
  18. N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36-41. https://doi.org/10.2307/2312781
  19. S. N. Maheshwari and R. Prasad, Some new separation axioms in bitopological spaces, Mat. Vesnik 12(27) (1975), no. 2, 159-162.
  20. S. N. Maheshwari and R. Prasad, Semi-open sets and semicontinuous functions in bitopological spaces, Math. Notae 26 (1977/78), 29-37.
  21. A. S. Mashhour, M. E. Abd El-Monsef, and S. N. El-Deeb, On precontinuous and weak precontinuous mappings, Proc. Math. Phys. Soc. Egypt No. 53 (1982), 47-53.
  22. M. N. Mukherjee and A. Debray, On S-cluster sets and S-closed spaces, Int. J. Math. Math. Sci. 23 (2000), no. 9, 597-603. https://doi.org/10.1155/S0161171200001277
  23. O. Njastad, On some classes of nearly open sets, Pacific J. Math. 15 (1965), 961-970. https://doi.org/10.2140/pjm.1965.15.961
  24. T. Noiri, A. Al-Omari, and M. S. M. Noorani, Weak forms of ${\omega}-open$ sets and decompositions of continuity, Eur. J. Pure Appl. Math. 2 (2009), no. 1, 73-84.
  25. T. Noiri, A. Al-Omari, and M. S. M. Noorani, Slightly ${\omega}-continuous$ functions, Fasc. Math. No. 41 (2009), 97-106.
  26. A. Al-Omari and M. S. M. Noorani, $Contra-{\omega}-continuous$ and almost $contra-{\omega}-continuous$, Int. J. Math. Math. Sci. 2007 (2007), Art. ID 40469, 13 pp.
  27. A. Al-Omari and M. S. M. Noorani, Regular generalized ${\omega}-closed$ sets, Int. J. Math. Math. Sci. 2007 (2007), Art. ID 16292, 11 pp.
  28. M. K. Singal and S. Prabha Arya, On almost-regular spaces, Glasnik Mat. Ser. III 4 (24) (1969), 89-99.
  29. T. Soundararajan, Weakly Hausdorff spaces and the cardinality of topological spaces, General Topology and its Relations to Modern Analysis and Algebra, III (Proc. Conf., Kanpur, 1968) pp. 301-306 Academia, Prague, 1971.
  30. W. J. Thron, Proximity structures and grills, Math. Ann. 206 (1973), 35-62. https://doi.org/10.1007/BF01431527
  31. N. Velicko, H-closed toplogical spaces, Mat. Sb. 70 (1966), 98-112
  32. N. Velicko, H-closed toplogical spaces, Amer. Math. Soc. Transl. 78 (1968), no. 2, 103-118.