DOI QR코드

DOI QR Code

SOME GENERALIZATIONS OF WEAKLY M-SEMI-CONTINUOUS AND WEAKLY M-PRECONTINUOUS FUNCTIONS

  • Received : 2015.07.09
  • Accepted : 2016.04.26
  • Published : 2016.05.15

Abstract

As a generalization of (i, j)-weakly m-continuous functions [43], we introduce the notion of weakly M(i, j)-continuous functions and obtain many characterizations and some properties of the functions. We show that the function is a unified form of some functions between m-spaces and certain kinds of weakly continuous functions in bitopological spaces.

Keywords

References

  1. M. E. Abd El-Monsef, S. N. El-Deeb, and R. A. Mahmoud, $\beta$-open sets and $\beta$-continuous mappings, Bull. Fac. Sci. Assiut Univ. 12 (1983), 77-90.
  2. M. E. Abd El-Monsef, R. A. Mahmoud, and E. R. Lashin, $\beta$-closure and $\beta$- interior, J. Fac. Ed. Ain Shams Univ. 10 (1986), 235-245.
  3. D. Andrijevic, Semi-preopen sets, Mat. Vesnik 38 (1986), 24-32.
  4. C. Boonpok, M-continuous functions on biminimal structure spaces, Far East J. Math. Sci. 43 (2010), 41-58.
  5. C. Boonpok, Biminimal structure spaces, Int. Math. Forum 5 (2010), no. 15, 703-707.
  6. C. Boonpok, Almost and weakly M-continuous functions on m-spaces, Far East J. Math. Sci. 43 (2010), 29-40.
  7. S. Bose and D. Sinha, Pairwise almost continuous map and weakly continuous map in bitopological spaces, Bull. Calcutta Math. Soc. 74 (1982), 195-206.
  8. C. Carpintero, R. Rosas, and M. Salas, Minimal structures and separation properties, Int. J. Pure Appl. Math. 34 (2007), no. 4, 473-488.
  9. S. G. Crossley and S. K. Hildebrand, Semi-closure, Texas J. Sci. 22 (1971), 99-112.
  10. 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 27(75) (1983), 311-315.
  11. E. Ekici, Generalization of weakly clopen and strongly $\theta$-b-continuous functions, Chaos, Solitons and Fractals 38 (2008), 79-88. https://doi.org/10.1016/j.chaos.2008.01.012
  12. E. Ekici, S. Jafari, M. Caldas, and T. Noiri, Weakly $\lambda$-continuous functions, Novi Sad J. Math. 38 (2008), no. 2, 47-56.
  13. E. Ekici and S. Jafari, On a new topology and decompositions of continuity, Int. J. Math., Game Theory Algebra 19 (1-2) (2010), 129-141.
  14. M. Jelic, A decomposition of pairwise continuity, J. Inst. Math. Comput. Sci. Math. Ser. 3 (1990), 25-29.
  15. M. Jelic, Feebly p-continuous mappings, Suppl. Rend. Circ. Mat. Palermo (2) 24 (1990), 387-395.
  16. A. Kar and P. Bhattacharyya, Bitopological preopen sets, precontinuity and preopen mappings, Indian J. Math. 34 (1992), 295-309.
  17. J. C. Kelly, Bitopological spaces, Proc. London Math. Soc. (3) 13 (1963), 71-89.
  18. F. H. Khedr, Weakly semi-continuous mappings in bitopological spaces, Bull. Fac. Sci. Assiut Univ. 21 (1992), 1-10.
  19. F. H. Khedr and S. M. Al-Areefi, Pairwise weakly and pairwise strongly irresolute (to appear).
  20. F. H. Khedr, S. M. Al-Areefi, and T. Noiri, Precontinuity and semi-precontinuity in bitopological spaces, Indian J. Pure Appl. Math. 23 (1992), 625-633.
  21. F. H. Khedr and T. Noiri, S-closed bitopological spaces, J. Egypt. Math. Soc. 15 (2007), 79-87.
  22. F. H. Khedr and T. Noiri, Almost s-continuous functions in bitopological spaces, J. Egypt. Math. Soc. 15 (2007), 89-99.
  23. N. Levine, A decomposition of continuity in topological spaces, Amer. Math. Monthly 68 (1961), 44-46. https://doi.org/10.2307/2311363
  24. N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36-41. https://doi.org/10.2307/2312781
  25. S. N. Maheshwari and R. Prasad, Semi open sets and semi continuous functions in bitopological spaces, Math. Notae 26 (1977/78), 29-37.
  26. H. Maki, K. C. Rao, and A. Nagor Gani, On generalizing semi-open and preopen sets, Pure Appl. Math. Sci. 49 (1999), 17-29.
  27. A. S. Mashhour, M. E. Abd El-Monsef, and S. N. El-Deep, On precontinuous and weak precontinuous mappings, Proc. Math. Phys. Soc. Egypt 53 (1982), 47-53.
  28. A. S. Mashhour, I. A. Hasanein, and S. N. El-Deeb, $\alpha$-continuous and $\alpha$-open mappings, Acta Math. Hungar. 41 (1983), 213-218. https://doi.org/10.1007/BF01961309
  29. W. K. Min, m-semiopen sets and M-semicontinuous functions on spaces with minimal structures, Honam Math. J. 31 (2009), 239-245. https://doi.org/10.5831/HMJ.2009.31.2.239
  30. W. K. Min, ${\alpha}m$-open sets and ${\alpha}M$-continuous functions, Commun. Korean Math. Soc. 25 (2010), 251-256. https://doi.org/10.4134/CKMS.2010.25.2.251
  31. W. K. Min and Y. K. Kim, m-preopen sets and M-precontinuity on spaces with minimal structures, Adv. Fuzzy Sets Systems 4 (2009), 237-245.
  32. W. K. Min and Y. K. Kim, On minimal precontinuous functions, J. Chungcheong Math. Soc. 22 (2009), 667-673.
  33. W. K. Min and Y. K. Kim, On weak M-semicontinuity on spaces with minimal structures, J. Chungcheong Math. Soc. 23 (2010), 223-229.
  34. W. K. Min and Y. K. Kim, On weak M-precontinuity on spaces with minimal structures, J. Chungcheong Math. Soc. 24 (2011), 11-17.
  35. M. N. Mukherjee, On pairwise S-closed bitopological spaces, Internat. J. Math. Math. Sci. 8 (1985), 729-745. https://doi.org/10.1155/S0161171285000825
  36. A. A. Nasef and T. Noiri, Feebly open sets and feeble continuity in bitopological spaces, Anal. Univ. Timisoara Ser. Mat. Inform. 36 (1998), 79-88.
  37. 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
  38. T. Noiri, The further unified theory for modifications of g-closed sets, Rend. Circ. Mat. Palermo 57 (2008), 411-421. https://doi.org/10.1007/s12215-008-0030-7
  39. T. Noiri and V. Popa, A new viewpoint in the study of irresoluteness forms in bitopological spaces, J. Math. Anal. Approx. Theory 1 (2006), 1-9.
  40. T. Noiri and V. Popa, A unified theory of weak continuity for multifunctions, Stud. Cerc. St. Ser. Mat. Univ. Bacau 16 (2006), 167-200.
  41. T. Noiri and V. Popa, A new viewpoint in the study of continuity forms in bitopological spaces, Kochi J. Math. 2 (2007), 95-106.
  42. T. Noiri and V. Popa, On weakly precontinuous functions in bitopological spaces, Soochow J. Math. 33 (2007), 87-100.
  43. T. Noiri and V. Popa, On some forms of weakly continuous functions in bitopological spaces, Demonstr. Math. 41 (2008), 685-700
  44. T. Noiri and V. Popa, On iterate minimal structures and iterate m-continuous functions, Anal. Sci. Budapest. 55 (2012), 67-79.
  45. T. Noiri and V. Popa, On iterate minimal structures and M-iterate continuous functions, Fasciculi Math. 50 (2013), 109-119.
  46. V. Popa and T. Noiri, Charactrizations of weakly quasicontinuous functions in bitopological spaces, Mathematica (Cluj) 39(62) (1997), 293-297.
  47. V. Popa and T. Noiri, On M-continuous functions, Anal. Univ. "Dunarea de Jos", Galati, Ser. Mat. Fis. Mec. Teor. 18 (23) (2000), no. 2, 31-41.
  48. V. Popa and T. Noiri, On the definitions of some generalized forms of continuity under minimal conditions, Mem. Fac. Sci. Kochi Univ. Ser. A Math. 22 (2001), 9-18.
  49. V. Popa and T. Noiri, A unified theory of weak continuity for functions, Rend. Circ. Mat. Palermo (2) 51 (2002), 439-464. https://doi.org/10.1007/BF02871853
  50. V. Popa and T. Noiri, On weakly m-continuous functions, Mathematica (Cluj) 45(68) (2003), 53-67.
  51. V. Popa and T. Noiri, Some properties of weakly quasicontinuous functions in bitopological spaces, Mathematica (Cluj) 46(69) (2004), 105-112.
  52. E. Rosas, N. Rajesh, and C. Carpintero, Some new types of open and closed sets in minimal structures, I, II, Int. Math. Forum 4 (2009), no. 44, 2169-2184, 2185-2198.