Pairwise Semiregular Properties on Generalized Pairwise Lindelöf Spaces

Main Article Content

Zabidin Salleh

Abstract

Let (X, τ1, τ2) be a bitopological space and (X, τs(1,2), τs(2,1)) its pairwise semiregularization. Then a bitopological property P is called pairwise semiregular provided that (X, τ1, τ2) has the property P if and only if (X, τs(1,2), τs(2,1)) has the same property. In this work we study pairwise semiregular property of (i, j)-nearly Lindelöf, pairwise nearly Lindelöf, (i, j)-almost Lindelöf, pairwise almost Lindelöf, (i, j)-weakly Lindelöf and pairwise weakly Lindelöf spaces. We prove that (i, j)-almost Lindelöf, pairwise almost Lindelöf, (i, j)-weakly Lindelöf and pairwise weakly Lindelöf are pairwise semiregular properties, on the contrary of each type of pairwise Lindelöf space which are not pairwise semiregular properties.

Article Details

References

  1. M.C. Datta, Projective Bitopological Spaces, J. Aust. Math. Soc. 13 (1972), 327–334. https://doi.org/10.1017/s1446788700013744.
  2. B.P. Dvalishvili, Bitopological Spaces: Theory, Relations With Generalized Algebraic Structures, and Applications, North-Holland Math. Stud. 199, Elsevier, 2005.
  3. A.J. Fawakhreh, A. Kılıçman, Semiregular Properties and Generalized Lindelöf Spaces, Mat. Vesnik. 56 (2004), 77-80.
  4. P. Fletcher, H.B. Hoyle, III, C.W. Patty, the Comparison of Topologies, Duke Math. J. 36 (1969), 325-331. https://doi.org/10.1215/s0012-7094-69-03641-2.
  5. A.A. Fora, H.Z. Hdeib, On Pairwise Lindelöf Spaces, Rev. Colombiana Mat. 17 (1983), 37-57.
  6. F.H. Khedr, A.M. Al-Shibani, On Pairwise Super Continuous Mappings in Bitopological Spaces, Int. J. Math. Math. Sci. 14 (1991), 715–722. https://doi.org/10.1155/s0161171291000960.
  7. A. Kılıçman, Z. Salleh, On Pairwise Lindelöf Bitopological Spaces, Topol. Appl. 154 (2007), 1600-1607. https://doi.org/10.1016/j.topol.2006.12.007.
  8. A. Kılıçman and Z. Salleh, Mappings and pairwise continuity on pairwise Lindelöf bitopological spaces, Albanian J. Math.1(2) (2007), 115-120.
  9. A. Kılıçman, Z. Salleh, Pairwise Almost Lindelöf Bitopological Spaces II, Malaysian J. Math. Sci. 1 (2007), 227-238.
  10. A. Kılıçman, Z. Salleh, Pairwise Weakly Regular-Lindelöf Spaces, Abstr. Appl. Anal. 2008 (2008), 184243. https://doi.org/10.1155/2008/184243.
  11. A. Kılıçman, Z. Salleh, On Pairwise Almost Regular-Lindelöf Spaces, Sci. Math. Japon. 70 (2009), 285-298.
  12. A. Kılıçman, Z. Salleh, Mappings and Decompositions of Pairwise Continuity on Pairwise Nearly Lindelöf spaces, Albanian J. Math. 4 (2010), 31-47.
  13. A. Kılıçman, Z. Salleh, On Pairwise Weakly Lindelöf Bitopological Spaces, Bull. Iran. Math. Soc. 39 (2013), 469- 486.
  14. M. Mršević, I. L. Reilly, M. K. Vamanamurthy, On Semi-Regularization Topologies, J. Aust. Math. Soc. A. 38 (1985), 40–54. https://doi.org/10.1017/s1446788700022588.
  15. M. Mršević, I.L. Reilly, M. K. Vamanamurthy, On Nearly Lindelöf Spaces, Glasnik Math. 21 (1986), 407-414.
  16. Z. Salleh, A. Kılıçman, Pairwise Nearly Lindelöf Bitopological Spaces, Far East J. Math. Sci. 77 (2013), 147-171.
  17. Z. Salleh, A. Kılıçman, Some Results on Pairwise Almost Lindelöf spaces, JP J. Geometry Topol. 15 (2014), 81-98.
  18. Z. Salleh, A. Kılıçman, Mappings and Decompositions of Pairwise Continuity on (i, j)-Almost Lindelöf and (i, j)-Weakly Lindelöf Spaces, Proyecciones J. Math. 40 (2021), 815-836. https://doi.org/10.22199/issn.0717-6279-3253.
  19. Z. Salleh and A. Kılıçman, Pairwise Semiregular Properties on Generalized Pairwise Regular-Lindelöf Spaces, Istanb. Univ., Sci. Fac., J. Math. Phys. Astron. 3 (2010), 127-136.
  20. A.R. Singal, S.P. Arya, On Pairwise Almost Regular Spaces, Glasnik Math. 6 (1971), 335-343.
  21. M.K. Singal, A.R. Singal, Some More Separation Axioms in Bitopological Spaces, Ann. Son. Sci. Bruxelles. 84 (1970), 207-230.
  22. L.A. Steen, J.A. Seebach Jr., Counterexamples in Topology, 2 nd Edition, Springer-Verlag, New York, 1978.
  23. J. Swart, Total Disconnectedness in Bitopological Spaces and Product Bitopological Spaces, Nederl. Akad. Wetensch., Proc. Ser. A 74 = Indag Math. 33 (1971), 135-145.