Pairwise Semiregular Properties on Generalized Pairwise Lindelöf Spaces

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DOI: https://doi.org/10.28924/2291-8639-21-2023-16
Published: Mar 3, 2023
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Zabidin Salleh, Pairwise Semiregular Properties on Generalized Pairwise Lindelöf Spaces, Int. J. Anal. Appl., 21 (2023), 16.

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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.

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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.